#d(y_, x_, n_) := If(n_>0 And ,  Block(p:=y_, For(j,1,n_,1,p:=d(p,x_)),p));


d(y_, x_,a_, n_) := If(isnumber(n),block(p:=d(y,x,n), replace(p,x,a)));
d(y_, xx_=a_, n_) := d(y, xx,a, n);

d(a_+b_,x_,n_):=d(a,x,n)+d(b,x,n);
d(a_=b_,x_,n_):=d(a,x,n)=d(b,x,n);

d(-y_,x_,n_):= -d(y,x,n);
d(d(y_,x_,n_),x_,m_) := d(y,x,n+m);
d(ds(y_,x_,n_),x_,m_) := ds(y,x,n+m);

d(a_*x_^m_,x_,n_):= If(isfree(a,x), a*d(x^m,x,n), 
  If(n>0 and isinteger(n), sum(binomial(n,r)*fallingfactorial(m,r)*d(a,x,n-r)*x^(m-r), r,0,n), 
  If(n>0 or (m>0 and isinteger(m)), sum(binomial(n,r)*fallingfactorial(m,r)*d(a,x,n-r)*x^(m-r), r,0,ceil(abs(m))) )));
d(x_^m_*y_,x_,n_):= If(isfree(y,x), y*d(x^m,x,n), 
  If(n>0 and isinteger(n), sum(binomial(n,r)*fallingfactorial(m,r)*d(y,x,n-r)*x^(m-r), r,0,n), 
  If(n>0 or (m>0 and isinteger(m)), sum(binomial(n,r)*fallingfactorial(m,r)*d(y,x,n-r)*x^(m-r), r,0,ceil(abs(m))) )));
d(a_*(c_+x_)^m_,x_,n_):= If(isfree(a,c,x), a*d((c+x)^m,x,n), 
  If(n>0 and isinteger(n), sum(binomial(n,r)*fallingfactorial(m,r)*d(a,x,n-r)*(c+x)^(m-r), r,0,n), 
  If(n>0 or (m>0 and isinteger(m)), sum(binomial(n,r)*fallingfactorial(m,r)*d(a,x,n-r)*(c+x)^(m-r), r,0,ceil(abs(m))) )));
d((c_+x_)^m_*y_,x_,n_):= If(isfree(y,c,x), y*d((c+x)^m,x,n), 
  If(n>0 and isinteger(n), sum(binomial(n,r)*fallingfactorial(m,r)*d(y,x,n-r)*(c+x)^(m-r), r,0,n), 
  If(n>0 or (m>0 and isinteger(m)), sum(binomial(n,r)*fallingfactorial(m,r)*d(y,x,n-r)*(c+x)^(m-r), r,0,ceil(abs(m))) )));
d(x_*y_,x_,n_):=If(isfree(y,x),If(n>1,0), If(n> -1,x*d(y,x,n)+n*d(y,x,n-1) ));
d(a_*x_,x_,n_):=If(isfree(a,x),If(n>1,0), x*d(a,x,n)+n*d(a,x,n-1));

d(e^(x_^m_)*x_,x_,n_):=If(n>= -1 and n<0 and m*n==(n-1), exp(x^m)*m^n);
d(e^(-x_^m_)*x_,x_,n_):=If(n>= -1 and n<0 and m*n==(n-1), exp(-x^m)*(-m)^n);

d(a_*b_*c_,x_,n_):= If(isfree(a,x), d(b*c,x,n)*a);

#d(a_*x_^m_,x_,n_):= If(isfree(a,x), If(m>0 and n>=m+1, 0, a*fallingfactorial(m,n)*x^(m-n)), If(isinteger(m), sum(binomial(n,r)*fallingfactorial(m,r)*d(a,x,n-r)*x^(m-r), r,0,abs(m),1) ));
#d(a_*(c_+x_)^m_,x_,n_):= If(isfree(c,x),If(isfree(a,x), If(m>0 and n>=m+1, 0, a*fallingfactorial(m,n)*(c+x)^(m-n)), If(isinteger(m), sum(binomial(n,r)*fallingfactorial(m,r)*d(a,x,n-r)*(c+x)^(m-r), r,0,abs(m),1) )));

d(a_^x_,x_,n_):= If(isfree(a,x), log(a)^n*a^x);
d(c_^(a_*x_),x_,n_) := If(isfree(c,a,x),(a*log(c))^n*c^(a*x));
d(c_^(x_+b_),x_,n_) := If(isfree(c,b,x), log(c)^n*c^(x+b));
d(c_^(b_+a_*x_),x_,n_) := If(isfree(c,b,a,x), (a*log(c))^n*c^(a*x+b));

d(e^f_,x_,p_):= If(p>=0 and p<=1, d(f,x)^p*e^f);
d(e^x_,x_,n_) := e^x;
d(e^(a_*x_),x_,n_) := If(isfree(a,x), a^n*e^(a*x));
d(e^(x_+b_),x_,n_) := If(isfree(b,x), e^(x+b));
d(e^(b_+a_*x_),x_,n_) := If(isfree(b,a,x), a^n*e^(a*x+b));

#d(x_^m_,x_,n_) := If(m==n, If(n>0, n!, (-1)^(1-n)/(-n-1)!*log(x)), fallingfactorial(m,n)*x^(m-n) );
#d((c_+x_)^m_,x_,n_) := If(isfree(c,x), If(m==n, If(n>0, n!, (-1)^(1-n)/(-n-1)!*log(c+x)), fallingfactorial(m,n)*(c+x)^(m-n) ));
#d((a_*x_)^m_,x_,n_) := If(isfree(a,x), If(m==n, If(n>0, a^m*n!, (-1)^(1-n)*a^m/(-n-1)!*log(x)), a^m*fallingfactorial(m,n)*x^(m-n) ));
#d((c_+a_*x_)^m_,x_,n_) := If(isfree(a,x) and isfree(c,x), If(m==n, If(n>0, n!*a^m, (-1)^(1-n)*a^m/(-n-1)!*log(c/a+x)), a^m*fallingfactorial(m,n)*(x+c/a)^(m-n) ));
#d(x_^m_,x_,n_) := If(isfree(m,x),If(m== -1 and n<0, (psi(1)-psi(-n)+log(x))*x^(-1-n)/(-1-n)!, m*fallingfactorial(m-1,n-1)*x^(m-n) ));

d(x_^m_,x_,n_) := If(isfree(m,x), If(m<0 and isinteger(m) and (m>=n or n<m+1), -(-1)^m/(-1-m)!/(m-n)!*x^(m-n)*log(x), fallingfactorial(m,n)*x^(m-n) ));
d((c_+x_)^m_,x_,n_) := If(isfree(c,m,x), If(m== -1 and n<0, log(c+x)*(c+x)^(-1-n)/(-1-n)!, fallingfactorial(m,n)*(c+x)^(m-n) ));
d((a_*x_)^m_,x_,n_) := If(isfree(a,m,x), If(m== -1 and n<0, log(x)*x^(-1-n)/(-1-n)!*a^m, fallingfactorial(m,n)*x^(m-n)*a^m ));
d((c_+a_*x_)^m_,x_,n_) := If(isfree(c,a,m,x), If(m== -1 and n<0, log(c+a*x)*(c+a*x)^(-1-n)/(-1-n)!*a^n, fallingfactorial(m,n)*(c+a*x)^(m-n)*a^n ));

#d((c_+x_)^m_,x_,n_) := If(isfree(c,x), If(m==n, If(m>0 and isinteger(m), m!, (-1)^(-m)/(-m-1)!*log(c+x)), If(m>n or m<0, fallingfactorial(m,n)*(c+x)^(m-n), m*d((c+x)^(m-1),x,n-1) )));
#d((a_*x_)^m_,x_,n_) := If(isfree(a,x), If(m==n, If(m>0 and isinteger(m), m!, (-1)^(-m)/(-m-1)!*log(x)*a^m), If(m>n or m<0, a^m*fallingfactorial(m,n)*x^(m-n), m*a^m*d(x^(m-1),x,n-1) )));
#d((c_+a_*x_)^m_,x_,n_) := If(isfree(a,x) and isfree(c,x), If(m==n, If(m>0 and isinteger(m), m!, (-1)^(-m)/(-m-1)!*log(x+c/a)*a^m), If(m>n or m<0, a^m*fallingfactorial(m,n)*(x+c/a)^(m-n), m*a^m*d((c/a+x)^(m-1),x,n-1) )));

#d(x_^k_,x_,k_):= If(isinteger(k) and k>0, k!, (-1)^(-k)/(-k-1)!*log(x));
#d((x_+c_)^k_,x_,k_):= If(isfree(c,x),If(isinteger(k) and k>0, k!, (-1)^(-k)/(-k-1)!*log(c+x) ));
#d((a_*x_)^k_,x_,k_):= If(isfree(a,x),If(isinteger(k) and k>0, k!, (-1)^(-k)/(-k-1)!*log(x)*a^k ));
#d((c_+a_*x_)^k_,x_,k_):= If(isfree(a,x) and isfree(c,x), If(isinteger(k) and k>0, k!, (-1)^(-k)/(-k-1)!*log(x+c/a)*a^k ));

d(sinh(x_)*f_^(x_),x_,n_):= If(isfree(f,x), e^((log(f)+1)*x)*(log(f)+1)^n/2-d(e^((log(f)-1)*x),x,n)/2 );
d(cosh(x_)*f_^(x_),x_,n_):= If(isfree(f,x), e^((log(f)+1)*x)*(log(f)+1)^n/2+d(e^((log(f)-1)*x),x,n)/2 );
d(sinh(x_)*f_^(-x_),x_,n_):= If(isfree(f,x), d(e^((1-log(f))*x),x,n)/2-e^((-log(f)-1)*x)*(-log(f)-1)^n/2 );
d(cosh(x_)*f_^(-x_),x_,n_):= If(isfree(f,x), d(e^((1-log(f))*x),x,n)/2+e^((-log(f)-1)*x)*(-log(f)-1)^n/2 );

d(sin(x_)*f_^(x_),x_,n_):=If(isfree(f,x), sin(x+n*pi/4)*f^(x)*2^(n/2)*log(f)^n);
d(cos(x_)*f_^(x_),x_,n_):=If(isfree(f,x), cos(x+n*pi/4)*f^(x)*2^(n/2)*log(f)^n);
d(sin(a_*x_)*e^(x_),x_,n_):=If(isfree(a,x), sin(a*x+n*atan(a))*e^(x)*(a^2+1)^(n/2));
d(cos(a_*x_)*e^(x_),x_,n_):=If(isfree(a,x), cos(a*x+n*atan(a))*e^(x)*(a^2+1)^(n/2));
d(sin(x_+b_)*e^(x_),x_,n_):=If(isfree(b,x), sin(x+n*pi/4+b)*e^(x)*2^(n/2));
d(cos(x_+b_)*e^(x_),x_,n_):=If(isfree(b,x), cos(x+n*pi/4+b)*e^(x)*2^(n/2));
d(sin(x_+b_)*e^(x_+c_),x_,n_):=If(isfree(b,c,x), sin(x+n*pi/4+b)*e^(x+c)*2^(n/2));
d(cos(x_+b_)*e^(x_+c_),x_,n_):=If(isfree(b,c,x), cos(x+n*pi/4+b)*e^(x+c)*2^(n/2));
d(sin(a_*x_)*e^(b_*x_),x_,n_):=If(isfree(a,b,x), sin(a*x+n*atan(a/b))*e^(b*x)*(a^2+b^2)^(n/2));
d(cos(a_*x_)*e^(b_*x_),x_,n_):=If(isfree(a,b,x), cos(a*x+n*atan(a/b))*e^(b*x)*(a^2+b^2)^(n/2));
d(sin(x_)*e^(b_*x_),x_,n_):=If(isfree(b,x), sin(x+n*atan(1/b))*e^(b*x)*(1+b^2)^(n/2));
d(cos(x_)*e^(b_*x_),x_,n_):=If(isfree(b,x), cos(x+n*atan(1/b))*e^(b*x)*(1+b^2)^(n/2));
d(sin(x_+c_)*e^(b_*x_),x_,n_):=If(isfree(c,b,x), sin(c+x+n*atan(1/b))*e^(b*x)*(1+b^2)^(n/2));
d(cos(x_+c_)*e^(b_*x_),x_,n_):=If(isfree(c,b,x), cos(c+x+n*atan(1/b))*e^(b*x)*(1+b^2)^(n/2));
d(e^(x_)*sin(a_*x_+b_),x_,n_):=If(isfree(a,b,x), sin(a*x+n*atan(a)+b)*e^(x)*(a^2+1)^(n/2) );
d(e^(x_)*cos(a_*x_+b_),x_,n_):=If(isfree(a,b,x), cos(a*x+n*atan(a)+b)*e^(x)*(a^2+1)^(n/2) );
d(e^(c_*x_)*sin(a_*x_+b_),x_,n_):=If(isfree(a,b,c,x), sin(a*x+b+n*atan(a/c))*e^(x)*(a^2+c^2)^(n/2) );
d(e^(c_*x_)*cos(a_*x_+b_),x_,n_):=If(isfree(a,b,c,x), cos(a*x+b+n*atan(a/c))*e^(x)*(a^2+c^2)^(n/2) );

#d(x_*(b_+x_)^k_,x_,n_):= If(isfree(b,x), sum(binomial(n,r)*fallingfactorial(k,n-r)*fallingfactorial(1,r)*(b+x)^(k-n+r)*x^(1-r),r,0,1) );
#d(x_*(b_+a_*x_)^k_,x_,n_):= If(isfree(a,x) and isfree(b,x), sum(binomial(n,r)*fallingfactorial(k,n-r)*fallingfactorial(1,r)*a^(r-n)*(b+a*x)^(k-n+r)*x^(1-r),r,0,1) );

#d(x_^k_*(b_+x_)^m_,x_,n_):= If(isfree(b,x), If(m>0 and (isinteger(m) or isinteger(n)), sum(binomial(k,r)*fallingfactorial(n,k-r)*fallingfactorial(m,r)*(x+b)^(n-k+r)*x^(m-r),r,0,m) ));
#d((x_+b_)^k_*(x_+d_)^m_,x_,n_):= If(isfree(b,x) and isfree(d,x), If(m>0 and (isinteger(m) or isinteger(n)), sum(binomial(n,r)*fallingfactorial(k,n-r)*fallingfactorial(m,r)*(x+b)^(k-n+r)*(x+d)^(m-r),r,0,m) ));
#d((a_*x_+b_)^k_*(c_*x_+d_)^m_,x_,n_):= If(isfree(a,x) and isfree(b,x) and isfree(c,x) and isfree(d,x), If(m>0 and (isinteger(m) or isinteger(n)), sum(binomial(n,r)*fallingfactorial(k,n-r)*fallingfactorial(m,r)*a^(r-n)*c^r*(a*x+b)^(k-n+r)*(c*x+d)^(m-r),r,0,m) ));
#d((c_+x_^2)^k_,x_,n_):=If(isfree(c,x), If(n>0 and n<1 and k<0, (-1)^(-n)*2^n*gamma(n-k)/gamma(-k)*(x^2+c)^(k-n)*x^(-2k-2n),	If(k>0, sum(binomial(n,r)*fallingfactorial(k,n-r)*fallingfactorial(k,r)*(x+sqrt(-c))^(k-n+r)*(x-sqrt(-c))^(k-r),r,0,max(k,1),1))));

d((x_+b_)^k_*(x_+d_)^m_,x_,n_):= If(isfree(b,k,d,m,x), sum(fallingfactorial(k,n-r)*fallingfactorial(m,r)*binomial(n,r)*(x+b)^(k-n+r)*(x+d)^(m-r),r,0,ceil(abs(m)),1));
d((a_*x_+b_)^k_*(c_*x_+d_)^m_,x_,n_):= If(isfree(a,b,k,c,d,m,x), sum(binomial(n,r)*a^(r-n)*c^r*fallingfactorial(k,n-r)*fallingfactorial(m,r)*(a*x+b)^(k-n+r)*(c*x+d)^(m-r),r,0,ceil(abs(m)),1));
d((x_+b_)^k_*(c_*x_+d_)^m_,x_,n_):= If(isfree(b,k,c,d,m,x), sum(binomial(n,r)*c^r*fallingfactorial(k,n-r)*fallingfactorial(m,r)*(x+b)^(k-n+r)*(c*x+d)^(m-r),r,0,ceil(abs(m)),1));
d(x_^k_*(x_+d_)^m_,x_,n_):= If(isfree(k,d,m,x), sum(binomial(n,r)*fallingfactorial(k,n-r)*fallingfactorial(m,r)*x^(k-n+r)*(x+d)^(m-r),r,0,ceil(abs(m)),1));
d(x_^k_*(c_*x_+d_)^m_,x_,n_):= If(isfree(k,c,d,m,x), sum(binomial(n,r)*c^r*fallingfactorial(k,n-r)*fallingfactorial(m,r)*x^(k-n+r)*(c*x+d)^(m-r),r,0,ceil(abs(m)),1));

d((c_+x_^2)^k_,x_,n_) := If(isfree(c,x), d((sqrt(-c)+x)^k*(-sqrt(-c)+x)^k,x,n));
d((c_-x_^2)^k_,x_,n_) := If(isfree(c,x), d((sqrt(c)+x)^k*(sqrt(c)-x)^k,x,n));
d(1/(k_+x_^2),x_,n_) := If(isfree(k,x), d(1/(x-sqrt(-k)),x,n)/2/sqrt(-k)-d(1/(x+sqrt(-k)),x,n)/2/sqrt(-k));
d(1/(k_-x_^2),x_,n_) := If(isfree(k,x), d(1/(sqrt(k)-x),x,n)/2/sqrt(k)-d(1/(x+sqrt(k)),x,n)/2/sqrt(k));
#d(1/(k_+(c_+x_)^2),x_,n_) := If(isfree(k,x) and isfree(c,x), d(1/(c+x-sqrt(-k)),x,n)/2/sqrt(-k)-d(1/(c+x+sqrt(-k)),x,n)/2/sqrt(-k));

#d(x_*(c_+x_^2)^k_,x_,n_):=If(n>0 and n<1 and k>-2 and k<0 and isfree(c,x), (-1)^n*2^n*gamma(n-k)/gamma(-k)*(x^2+c)^(k-n)*x^(-k-n));

#d(log(x_)*x_^k_,x_,n_) := If(k>0, If(n>k, -(-1)^(n-k)*k!/(n-k-1)!*x^(k-n), If(k>n, k!/(k-n)!*x^(k-n)*(psi(k+1)-psi(k-n+1)+log(x)))), If(k<0, (-1)^n*gamma(n-k)/gamma(-k)*x^(k-n)*(psi(-k+1)-psi(-k-n+1)+log(x))));
#d(log(x_)*x_^m_,x_,n_) := If(n>m and m>0, -sum((-1)^(n-r)*binomial(n,r)*gamma(n-r)*fallingfactorial(m,r)*x^(m-n),r,0,n-1,1), fallingfactorial(m,n)*x^(m-n)*(psi(m+1)-psi(m-n+1)+log(x)));
#d(log(x_)*x_^m_,x_,n_) := If(n>m and m>0, -(-1)^(n-m)*m!*(n-m-1)!*x^(m-n), If(not(isinteger(n)), sum(binomial(n,r)*fallingfactorial(m,r)*(log(x)-psi(1+r-n)-gamma)/(r-n)!,r,0, abs(m),1)*x^(m-n) ));
#d(log(x_)*x_^m_,x_,n_) := If(n>m and isinteger(m-n), -(-1)^(n-m)*m!*(n-m-1)!*x^(m-n), If((isinteger(m) or isinteger(n-m)) and n>0 and n<1 and n<>m, fallingfactorial(m,n)*log(x)*x^(m-n)+fallingfactorial(m,n)*n/ceil(m-n)*x^(m-n), fallingfactorial(m,n)*(psi(1+m)-psi(1+m-n)+log(x))*x^(m-n)));
#d(log(c_+x_)*x_^m_,x_,n_) := If(n>m+1 and m>0, -(-1)^(n-m)*m!*(n-m-1)!*x^(m-n), If(n>0 and n<1 and m<= -1 and m-n <= -1, fallingfactorial(m,n)*log(c+x)*x^(m-n)+fallingfactorial(m,n)*n/ceil(m-n)*x^(m-n), fallingfactorial(m,n)*(psi(1+m)-psi(1+m-n)+log(c+x))*x^(m-n)));
#d(log(x_)*x_^m_,x_,n_) := If(n>m+1 and m>0, -(-1)^(m-n)*m!*(n-m-1)!*x^(m-n),  fallingfactorial(m,n)*(psi(1+abs(m))-psi(1+abs(m)-n)+log(x))*x^(m-n) );

d(x_*log(x_),x_,n_) := If(n>1, (-1)^n*(n-2)!*x^(1-n),  (psi(2)-psi(2-n)+log(x))/gamma(2-n)*x^(1-n));

#d(x_*log(a_*x_),x_,n_) := If(isfree(a,x), (If(n>1, (-1)^n*(n-2)!*x^(1-n), If(n<1, 1/(1-n)!*x^(1-n)*(psi(2)-psi(2-n))+log(a*x)/(1-n)!*x^(1-n)))));
#d(x_*log(x_+c_),x_,n_) := If(isfree(c,x), If(n>1, (-1)^n*((n-2)!*(x+c)^(1-n)+c*gamma(n)*(x+c)^(-n)), If(n<1, (-1)^n*c*gamma(n)*(x+c)^(-n)+1/(1-n)!*(x+c)^(1-n)*(psi(2)-psi(2-n))+log(x+c)/(1-n)!*(c+x)^(1-n))));

d(x_^m_*log(x_),x_,n_) := If(isinteger(n-m) and n>m, -(-1)^(n-m)*(n-m-1)!*gamma(1+m)*x^(m-n), If(m> -1, fallingfactorial(m,n)*(psi(1+m)-psi(1+m-n)+log(x))*x^(m-n), 
  If(m== -1 and n>= -1 and n<=0, x^(-1-n)*log(x)^(1-n)/gamma(2-n), If(m< -1, fallingfactorial(m,n)*(psi(abs(m))-psi(abs(m-n))+log(x))*x^(m-n) ))));

#d(log(x_)*x_^m_,x_,n_) := If(n>m and m>0, m*d(log(x)*x^(m-1),x,n-1)+fallingfactorial(m-1,n-1)*x^(m-n), If(m>0, fallingfactorial(m,n)*(psi(1+m)-psi(1+m-n)+log(x))*x^(m-n), 
  If(m<0, fallingfactorial(m,n)*(psi(abs(m))-psi(abs(m-n))+log(x))*x^(m-n) )));
#d(log(x_)*x_^m_,x_,n_) := If(n>m and m>0, -(-1)^(m-n)*m!*(n-m-1)!*x^(m-n),  If(m>0, fallingfactorial(m,n)*(psi(1+m)-psi(1+m-n)+log(x))*x^(m-n), fallingfactorial(m,n)*(psi(abs(m))-psi(abs(m-n))+log(x))*x^(m-n) ));
#d(log(x_)*x_^m_,x_,n_) := If(n>m+1 and m>0, -(-1)^(m-n)*m!*(n-m-1)!*x^(m-n), If(m>0 or m==n, fallingfactorial(m,n)*(psi(1+m)-psi(1+m-n)+log(x))*x^(m-n), fallingfactorial(m,n)*(psi(abs(m))-psi(abs(m-n))+log(x))*x^(m-n) ));
#d(log(x_)*x_^m_,x_,n_) := If(n>m+1 and m>0, -(-1)^(m-n)*m!*(n-m-1)!*x^(m-n), If(m>0, m*d(log(x)*x^(m-1),x,n-1)+d(x^(m-1),x,n-1), fallingfactorial(m,n)*(psi(abs(m))-psi(abs(m-n))+log(x))*x^(m-n) ));
#d(log(x_)*x_^m_,x_,n_) := If(n>m+1 and m>0, -(-1)^(m-n)*m!*(n-m-1)!*x^(m-n), fallingfactorial(m,n)*(psi(abs(m))-psi(abs(m-n))+log(x))*x^(m-n) );

#d(log(x_)^m_,x_,p_):= If(p>= -1 and p<=0, (-1)^m*d(gamma(m-p, -log(x)),x,1+p), If(p>=0 and p<=1, fallingfactorial(m,p)*x^(-p)*log(x)^(m-p) ));
d(log(x_)^m_,x_,p_):=  If(p>=0 and p<=1, fallingfactorial(m,p)*x^(-p)*log(x)^(m-p) );
d(log(x_+c_)^m_,x_,p_):=  If(p>=0 and p<=1 and isfree(c,x), fallingfactorial(m,p)*(x+c)^(-p)*log(x+c)^(m-p) );
#d(x_^n_*log(x_)^m_,x_,p_):= d(integrate(x^n*log(x)^m,x),x,1+p);
d(x_^n_*log(x_)^m_,x_,p_):= If(p>=0 and p<=1 and n==p-1, fallingfactorial(m,p)*log(x)^(m+p-1));
d(x_^p_*log(x_)^m_,x_,p_):= If(p>= -1 and p<=0, fallingfactorial(m,p)*log(x)^(m-p) );
d(log(x_)^m_/x_,x_,p_):= If(p>= -1 and p<=0, fallingfactorial(m,p)*x^(-1-p)*log(x)^(m-p) );

d(e^x_*x_^m_,x_,n_):= If(n>0 and isinteger(n), exp(x)*sum(binomial(n,r)*fallingfactorial(m,r)*x^(m-r), r,0,n), 
  If(n>0 or (m>0 and isinteger(m)), exp(x)*sum(binomial(n,r)*fallingfactorial(m,r)*x^(m-r), r,0,ceil(abs(m))) ));
d(e^(a_*x_)*x_^m_,x_,n_):= If(isfree(m,a,x), If(n>0 and isinteger(n), exp(a*x)*sum(a^(n-r)*binomial(n,r)*fallingfactorial(m,r)*x^(m-r), r,0,n), 
  If(n>0 or (m>0 and isinteger(m)), exp(a*x)*sum(a^(n-r)*binomial(n,r)*fallingfactorial(m,r)*x^(m-r), r,0,ceil(abs(m))) )));
d(e^(c_+a_*x_)*x_^m_,x_,n_):= If(isfree(m,c,a,x), If(n>0 and isinteger(n), exp(c+a*x)*sum(a^(n-r)*binomial(n,r)*fallingfactorial(m,r)*x^(m-r), r,0,n), 
   If(n>0 and (m>0 and isinteger(n)), exp(c+a*x)*sum(a^(n-r)*binomial(n,r)*fallingfactorial(m,r)*x^(m-r), r,0,ceil(abs(m))) )));
d(e^(c_+x_)*x_^m_,x_,n_):= If(isfree(m,c,x), If(n>0 and isinteger(n), exp(c+x)*sum(binomial(n,r)*fallingfactorial(m,r)*x^(m-r), r,0,n), 
   If(n>0 or (m>0 and isinteger(m)), exp(c+x)*sum(binomial(n,r)*fallingfactorial(m,r)*x^(m-r), r,0,ceil(abs(m))) )));
d(e^x_*(c_+x_)^m_,x_,n_):= If(isfree(c,m,x), If(n>0 and isinteger(n), exp(x)*sum(binomial(n,r)*fallingfactorial(m,r)*(c+x)^(m-r), r,0,n), 
   If(n>0 or (m>0 and isinteger(m)), exp(x)*sum(binomial(n,r)*fallingfactorial(m,r)*(c+x)^(m-r), r,0,ceil(abs(m))) )));
 
d(x_^m_*sin(x_),x_,p_):= sum(binomial(p,r)*fallingfactorial(m,r)*x^(m-r)*sin(x+(p-r)*pi/2), r,0,abs(m),1);
d(x_^m_*cos(x_),x_,p_):= sum(binomial(p,r)*fallingfactorial(m,r)*x^(m-r)*cos(x+(p-r)*pi/2), r,0,abs(m),1);

d(x_^m_*sinh(x_),x_,p_):= sum(binomial(p,r)*fallingfactorial(m,r)*x^(m-r)*(exp(x)-(-1)^(p-r)*exp(-x))/2, r,0,abs(m),1);
d(x_^m_*cosh(x_),x_,p_):= sum(binomial(p,r)*fallingfactorial(m,r)*x^(m-r)*(exp(x)+(-1)^(p-r)*exp(-x))/2, r,0,abs(m),1);

#d(x_^m_*e^(x_+c_),x_,n_):= If(isfree(c,x), exp(x+c)*sum(binomial(n,r)*fallingfactorial(m,r)*x^(m-r), r,0,If(m>0,m,n),1));
#d(x_^m_*e^(a_*x_),x_,n_):= If(isfree(a,x), a^(n-m)*exp(a*x)*sum(a^(m-r)*binomial(n,r)*fallingfactorial(m,r)*x^(m-r), r,0,If(m>0,m,n),1));
#d(x_^m_*e^(a_*x_+c_),x_,n_):= If(isfree(a,x) and isfree(c,x), a^(n-m)*exp(a*x+c)*sum(a^(m-r)*binomial(n,r)*fallingfactorial(m,r)*x^(m-r), r,0,If(m>0,m,n),1));
#d(x_^m_*e^(-x_),x_,n_):= (-1)^(-n)*exp(-x)*sum((-1)^r*binomial(n,r)*fallingfactorial(m,r)*x^(m-r), r,0,abs(m),1);
#d((c_+x_)^m_*e^x_,x_,n_):= If(isfree(c,x), exp(x)*sum(binomial(n,r)*fallingfactorial(m,r)*(c+x)^(m-r), r,0,If(m>0,m,round(n)),1));

#d(x_^m_*e^(a_*x_),x_,p_):= If(p<0 and p>= -1, gamma(m-p,-a*x)*(-1)^(1-m)/a^(m+p+1));
#d(x_^m_*e^(a_*x_),x_,p_):= If(p<0 and p>= -1, gamma(m-p,-a*x)*(-1)^(p+1)/a/(-a)^(m+p));
#d(x_^n_*e^(a_*x_),x_,p_):= If(p>=-1 and isfree(a,x),(-a)^n*d(gamma(n+abs(p),-a*x),x,p+1));
#d(x_^m_*e^(-x_),x_,p_):= If(p<0 and p>= -1, (-1)^(p)*gamma(m-p,x));
#d(x_^m_*e^(x_),x_,p_):= If(p<0 and p>= -1, (-1)^(1-m)*gamma(m-p,-x));

d(e^(x_^m_),x_,p_):= If(p>= -1 and p<=0, (-1)^(p/m-p)*m^p*gamma(1+p-p/m, -x^m), If(p>=0 and p<=1, (m*x^(m-1))^p*e^(x^m) ));
d(e^(a_*x_^m_),x_,p_):= If(p>= -1 and p<=0, (-a)^(p/m-p)*(a*m)^p*gamma(1+p-p/m, -a*x^m), If(p>=0 and p<=1, (a*m*x^(m-1))^p*e^(a*x^m) ));
#d(e^(x_^m_)*x_^n_,x_,p_):= If(m== -1, (-1)^p*exp(1/x)*x^(n-2p), d(integrate(e^(x^m)*x^n,x),x,1+p) );
#d(e^(-x_^m_)*x_^n_,x_,p_):= If(m== -1, (-1)^(2p)*exp(-1/x)*x^(n-2p),If(p>= -1 and p<=0, d(integrate(e^(-x^m)*x^n,x),x,1+p) ));
d(e^(x_^m_)*x_^n_,x_,p_):= If(p>= -1 and p<0, If(n==p*(1-m), exp(x^m)*m^p, -(-1)^((n-p)/(p*m))*m^p*gamma((n-p)/(-p*m),-x^m) ));
d(e^(a_*x_^m_)*x_^n_,x_,p_):= If(p>= -1 and p<0 and n==p*(1-m), exp(a*x^m)*(a*m)^p);

#d(e^(1/x_)*x_^n_,x_,p_):=  (-1)^p*exp(1/x)*x^(n-2p) ;

#d(ei(m_, x_), x_,p_) := If(p>0 and p<=1, exp(x)*x^(m-p), If(p> -1 and p<=0, ei(m-p,x), If(p> -2 and p<=-1, -(-1)^p*x*ei(m,x)+(-1)^p*ei(m-p,x) )));

#d(x_^d_*mittag(a_,b_,x_^a_),x_,n_) := If(d==b-1, x^(d-n)*mittag(a,b-n,x^a) );

d(x_^d_*mittag(a_,b_,x_^a_),x_,a2_) := If(d==b-1, If(a2==a,x^(b-a-1)/gamma(b-a)+x^d*mittag(a,b,x^a),If(a2== -a,x^(a+b-1)*mittag(a,a+b,x^a),x^(d-n)*mittag(a,b-n,x^a) )));
d(x_^d_*mittag(a_,b_,c_*x_^a_),x_,a2_) := If(isfree(c,x) and d==b-1, If(a2==a,x^(b-a-1)/gamma(b-a)+c*x^d*mittag(a,b,c*x^a), If(a2== -a,x^(a+b-1)*mittag(a,a+b,c*x^a),c*x^(d-n)*mittag(a,b-n,c*x^a) )));
d(mittag(a_,a_,c_*x_^a2_),x_,a_) := If(isfree(c,x) and a2== -a, c*x^(-2a)*mittag(a,a,c*x^a2) );
d(mittag(a_,a_,x_^a2_),x_,a_) := If( a2== -a, x^(-2a)*mittag(a,a,x^a2) );
d(mittag(a_,b_,x_^a_),x_,a_) :=  1/x^a/gamma(b-a)+mittag(a,b,x^a);
d(mittag(a_,b_,c_*x_^a_),x_,a_) := If(isfree(c,x), 1/x^a/gamma(b-a)+c*mittag(a,b,c*x^a) );

d(mittag(p_,f_),x_,p_):=  d(f,x,p)*mittag(p,f) ;

#d(mittag(p_,c_*f_),x_,p_):= If(p>=0 and p<=1 and isfree(c,x), c*d(f,x,p)*mittag(p,c*f) );
#d(mittag(p_,c_*x_),x_,p_):= If(p>=0 and p<=1 and isfree(c,x), c*x^(1-p)*mittag(p,c*x)/(1-p)! );
#d(mittag(a_,x_^a_),x_,a2_) := If(a2==a, mittag(a,x^a), If(a2== -a, mittag(a,x^a)+1 ));
#d(mittag(a_,c_*x_^a_),x_,a2_) := If(a2== -a and isfree(c,x), 1/c*mittag(a,c*x^a)+1 );
#d(mittag(a_,c_*x_^a_),x_,a_) := If(isfree(c,x), c*mittag(a,c*x^a) );

d(gamma(n_,a_*log(x_)),x_,p_) := If(p>= -1 and p<=0, d(log(x)^n/n+x^(-p)*gamma(n, a*log(x)),x,1+p), If(p>=0 and p<=1,  fallingfactorial(n,p)*(-a^n)*x^(-p-a)*log(x)^(n-p)/n ));
d(gamma(n_,log(x_)),x_,p_) := If(p>= -1 and p<=0, d(log(x)^n/n+x^(-p)*gamma(n, log(x)),x,1+p), If(p>=0 and p<=1,  (-1)^p*x^(-2p)*log(x)^(n-p) ));
d(gamma(n_,x_^m_),x_,p_):= If(p>= -1 and p<=0, (-m)^p*gamma(n+p-p/m,x^m) );

#d(gamma(n_,x_^m_),x_,p_):= If(p> -1 and p<=0, (-m)^p*gamma(((m-1)*(1+p)+(n-1)*m+1)/m,x^m), If(p>0 and p<=1, (-m)^p*x^((m-1)*p+(n-1)*m)*e^(-x^m) ));
#d(gamma(n_,-x_^m_),x_,p_):= If(p> -1 and p<=0, -m^p*(-1)^(-((m-1)*(1+p)+(n-1)*m+1)/m)*gamma(((m-1)*(1+p)+(n-1)*m+1)/m,-x^m), If(p>0 and p<=1, (-1)^(n-1)*m^p*x^((m-1)*p+(n-1)*m)*e^(x^m) ));
#d(gamma(n_,-x_^m_),x_,p_):=  If(p>0 and p<=1, (-1)^(n-1)*m^p*x^((m-1)*p+(n-1)*m)*e^(x^m) );
#d(gamma(n_,a_*x_^m_),x_,p_):= If(abs(p)<= 1 and isfree(a,x), (-m)^p*gamma(n+p-p/m,a*x^m) );
#d(gamma(n_,-x_^m_),x_,p_):= If(p>= -1 and p<=0, -(-1)^n*m^p*gamma(n+p-p/m, -x^m) );

d(gamma(n_,a_*x_^m_),x_,p_):=  If(p>0 and p<=1 and isfree(a,x) and n+p-p/m==1, (a)^(p/m-p)*(-a*m)^p*e^(-a*x^m) );

#d(gamma(m_, x_), x_,p_) := If(p>0 and p<=1, (-1)^(p)*exp(-x)*x^(m-p), If(p> -1 and p<=0, (-1)^(p)*gamma(m-p,x), If(p> -2 and p<=-1, -(-1)^p*x*gamma(m,x)+(-1)^p*gamma(m-p,x) )));
#d(gamma(m_, x_), x_,p_) := If(p>0 and p<=1, (-1)^(p)*exp(-x)*x^(m-p), If(p>= -1 and p<0, -(-1)^p*x*gamma(m,x)+(-1)^p*gamma(m-p,x) ));
#d(gamma(m_, x_), x_,p_) := If(p== -1, x*gamma(m,x)-gamma(m+1,x), If(m>0 and p>0 and p<1, sum((-1)^(p+r)*binomial(p,r)*fallingfactorial(m-1,r)*gamma(m-r,x), r,0,ceil(abs(m-1))), 
  If(not(isinteger(p)) and isnumber(p), d(d(gamma(m,x),x,p-floor(p)),x, floor(p)) )));
#d(gamma(n_,x_),x_,p_):= If(p>= -1 and p<0, x*gamma(n,x)+p*gamma(n+1,x), If(p>0 and p<=1, (-1)^p*x^(n-1)*e^(-x) ));
#d(gamma(m_, x_), x_,-1) := x*gamma(m,x)-gamma(m+1,x);

d(gamma(m_, x_), x_,p_) := If(p== -1, x*gamma(m,x)-gamma(m+1,x), If( p>0 and p<1, sum((-1)^(p+r)*binomial(p,r)*fallingfactorial(m-1,r)*gamma(m-r,x), r,0,ceil(abs(m-1))) ));

#d(gamma(n_, x_), x_,p_) := If( p>=0 and p<=1, -d( e^(-x)*x^(-1+n),x, p-1));
#d(gamma(n_, x_), x_,p_) := If(p>= -1 and p<=0, x^(-p)*gamma(n,x)+(-1)^p*gamma(n-p,x) );

d(zeta(x_),x_,p_):= gamma(p)*zeta(p,x);
d(zeta(m_,x_),x_,p_):= risingfactorial(m,p)*zeta(m+p,x);
d(ei(x_),x_,p_):=If(p>=0, -(-1)^p*gamma(p,-x)/x^p);
d(ei(m_, x_), x_,p_) := (-1)^p*ei(m-p,x);
d(erf(x_),x_,p_):= If(abs(p)<= 1, -(-2)^p*gamma((p+1)/2,x^2)/sqrt(pi) );
d(erfc(m_, x_), x_,p_) := erfc(m-p,x);
d(erfi(x_),x_,p_):= If(abs(p)<= 1, (-2)^p*gamma((p+1)/2,-x^2)/sqrt(pi) );
d(psi(m_,x_), x_,p_) := psi(m+p,x);
d(psi(x_),x_,p_) := psi(p,x);

d(loggamma(x_),x_,p_) := psi(p-1,x);
d(gamma(x_),x_,p_) := If(p>=0 and p<=1, gamma(x)*(psi(p-1,x)*p+1-p));
d(sin(x_),x_,n_) := sin(x+n*pi/2);
d(cos(x_),x_,n_) := cos(x+n*pi/2);
d(sinh(x_),x_,n_) := If(iseven(n), sinh(x), If(isodd(n), cosh(x), exp(x)/2-exp(-x)*(-1)^n/2));
d(cosh(x_),x_,n_) := If(iseven(n), cosh(x), If(isodd(n), sinh(x), exp(x)/2+(-1)^n*exp(-x)/2));
#d(sinh(x_),x_,n_) := If(iseven(n), sinh(x), If(isodd(n), cosh(x), sinh(x-n*pi/2*i)*(-1)^(n/2) ));
#d(cosh(x_),x_,n_) := If(iseven(n), cosh(x), If(isodd(n), sinh(x), cosh(x-n*pi/2*i)*(-1)^(n/2) ));
d(abs(y_),x_,n_) := d(y,x,n)/sgn(y);
d(abs(x_),x_,n_) :=  If(n>1,0, If(n<1, 1/sgn(x)/(1-n)!*x^(1-n) ));
d(sgn(y_),x_,n_) := If(n>0,0, If(n<0, sgn(y)*x^(-n)/(-n)! ));
d(csgn(y_),x_,n_) := If(n>0,0, If(n<0, csgn(y)*x^(-n)/(-n)! ));
d(delta(y_),x_,n_):=If(n>0,0, If(n<0, theta(y)*x^(-n)/(-n)! ));
d(log(x_),x_,n_):= If(n>0 and isinteger(n), -(-1)^n*(n-1)!/x^n,If(abs(n)<=1,(log(x)+psi(1)-psi(1-n))/gamma(1-n)/x^n ));

#d(log(x_),x_,n_):= If(n>1, -(-1)^n*(n-1)!/x^n, (-1)^(n+1)*(n-1)!*(1+log(x))*x^(-n));
#d(log(x_),x_,n_):= d(1/x,x,n-1);
#d(atan(x_),x_,n_):= If(n>0 and n<1, (-1)^(n-1)*2^(n-1)*(n-1)!*(1+x^2)^(-n));
#d(acot(x_),x_,n_):= If(n>0 and n<1, (-1)^n*2^(n-1)*(n-1)!*(1+x^2)^(-n));
#d(atanh(x_),x_,n_):= If(n>0 and n<1, 2^(n-1)*(n-1)!*(-1)^n*(-1+x^2)^(-n));
#d(acoth(x_),x_,n_):= If(n>0 and n<1, d(log(x+1),x,n)/2-d(log(x-1),x,n)/2);
#d(acoth(x_),x_,n_):=  d(1/(x+1),x,n-1)/2-d(1/(x-1),x,n-1)/2;
#d(atanh(x_),x_,n_):=  d(1/(x+1),x,n-1)/2-d(1/(1-x),x,n-1)/2;
#d(atan(x_),x_,n_):=   d(1/(-x+i),x,n-1)/2-d(1/(i+x),x,n-1)/2;
#d(acot(x_),x_,n_):=   d(1/(x-i),x,n-1)/2-d(1/(i+x),x,n-1)/2;

d(atanh(x_),x_,n_):= (psi(1)-psi(-n)+log(x+1))*(x+1)^(-n)/(-n)!/2+(psi(1)-psi(-n)+log(1-x))*(1-x)^(-n)/(-n)!/2;
d(acoth(x_),x_,n_):= (psi(1)-psi(-n)+log(x+1))*(x+1)^(-n)/(-n)!/2+(psi(1)-psi(-n)+log(1-x))*(1-x)^(-n)/(-n)!/2;
d(atan(x_),x_,n_):= (psi(1)-psi(-n)+log(x-i))*(x-i)^(-n)/(-n)!/2-(psi(1)-psi(-n)+log(i+x))*(i+x)^(-n)/(-n)!/2;
d(acot(x_),x_,n_):= (psi(1)-psi(-n)+log(x+i))*(x+i)^(-n)/(-n)!/2-(psi(1)-psi(-n)+log(x-i))*(x-i)^(-n)/(-n)!/2;
d(sqrt(x_),x_, n_):= If(n==0.5, i/(-1.5)!*log(x), (1/2)!/(1/2-n)!*x^(1/2-n));
d(exp(x_),x_,n_) := exp(x);

d(1/sqrt(x_),x_,n_):= If(n==-0.5, sqrt(pi)/2*log(x), (-1)^(-n)*gamma(n+1/2)/gamma(1/2)*x^(-1/2-n));

d(x_,x_,n_) :=  If(n>1, 0, x^(1-n)/(1-n)!);

d(y_,x_,1) := d(y,x);
d(y_,x_,0) := y;

d(y_,xx_=a_) := block(p:=d(y,xx), replace(p,xx,a));

d(sum(f(x_),x_,a_, b_)) := sum(d(f(x)),x,a,b);
d(infsums(x_^k_/k_!,k_),x_,n_) := infsums(x^k/k!,k);
d(infsums(x_^k_/k_!,k_),x_) := infsums(x^k/k!,k);
d(infsums(x_^(2k_)/(2k_)!,k_),x_) := infsums(x^(2k+1)/(2k+1)!,k);
d(infsums(x_^(1+2k_)*1/(1+2k_)!,k_),x_) := infsums(x^(2k)/(2k)!,k);
d(infsums((-1)^k_*x_^(2k_)*1/(2k_)!,k_),x_) := -infsums((-1)^k*x^(2k+1)/(2k+1)!,k);
d(infsums((-1)^k_*x_^(1+2k_)*1/(1+2k_)!,k_),x_) := infsums((-1)^k*x^(2k)/(2k)!,k);

d(mittag(a_,b_, x_), x_) := (mittag(a,b-1,x)-(b-1)*mittag(a,b,x))/(a*x);
d(mittag(a_,x_), x_) := mittag(a,0,x)/a/x;
d(gamma(n_, x_), x_) := -x^(n-1)*exp(-x);
d(zeta(n_,x_), x_) := n*zeta(1+n,x);
d(psi(n_,x_), x_) := psi(1+n,x);
d(ei(n_, x_), x_) := -ei(n-1,x);
d(erf(n_, x_), x_) := n!/sqrt(pi)*exp(-x^n);
d(erfc(n_, x_), x_) := erfc(n-1,x);
d(li(n_,x_), x_) := li(n-1,x)/x;
d(log(n_,x_), x_) := 1/x/log(n) ;

d(d(y_,x_,n_),x_) := d(y,x,n+1);
d(d(y_,x_),x_) := d(y,x,2);

#d(integrate(y_,x_),t_) := If(not(isfree(y,t)),integrate(d(y,t),x));
d(integrate(y_,x_),x_) := y;
d(integrate(y_,t_,a_,b_),x_) := If(not(isfree(a,x)), -replace(y,t,a)*d(a,x), If(not(isfree(b,x)), replace(y,t,b)*d(b,x) ));
d(integrates(y_,t_,a_,b_),x_) := If(not(isfree(a,x)), -replace(y,t,a)*d(a,x), If(not(isfree(b,x)), replace(y,t,b)*d(b,x) ));

d(sgn(x_),x_) := 0;
d(csgn(x_),x_) := 0;
d(theta(x_),x_) := 0;
d(delta(x_),x_) := 0;

d(log10(x_),x_) := 1/x/log(10);

d(atan(x_,y_),y_) := y/(x^2+y^2);
d(atan(x_,y_),x_) := -y/(x^2+y^2);
d(acot(x_),x_) := -1/(x^2+1);
d(asec(x_),x_) := 1/(x*sqrt(1-x^2));
d(acsc(x_),x_) := -1/(x*sqrt(1+x^2));

d(tanh(x_),x_) := sech(x)^2;
d(coth(x_),x_) := -csch(x)^2;
d(sech(x_),x_) := -tanh(x)*sech(x);
d(csch(x_),x_) := -coth(x)*csch(x);

d(asin(x_),x_) := 1/(1-x^2)^(1/2);
d(acos(x_),x_) := -1/sqrt(1-x^2);
d(asinh(x_),x_) := 1/sqrt(x^2+1);
d(acosh(x_),x_) := 1/sqrt(x^2-1);
d(atanh(x_),x_) := 1/(1-x^2);
d(acoth(x_),x_) := 1/(1-x^2);
d(asech(x_),x_) := -1/(x*sqrt(1-x^2));
d(acsch(x_),x_) := -1/(x*sqrt(1+x^2));

d(gauss(x_), x_) := -e^(-x^2/2)/sqrt(2pi)*x;
d(erf(x_), x_) := 2/sqrt(pi)*exp(-x^2);
d(erfi(x_), x_) := 2/sqrt(pi)*exp(x^2);
d(gamma(x_), x_) := gamma(x)*psi(x);
d(loggamma(x_), x_) := psi(x);
d(zeta(x_), x_) := zeta(1,x);
d(psi(x_), x_) := psi(1,x);
d(factorial(x_), x_) := gamma(x+1)*psi(x+1);
d(li(x_),x_) := 1/log(x);
d(ei(x_),x_) := exp(x)/x;
d(si(x_),x_) := sin(x)/x;
d(ci(x_),x_) := cos(x)/x;
d(shi(x_),x_) := sinh(x)/x;
d(chi(x_),x_) := cosh(x)/x;
d(sqrt(x_),x_):= 1/2*x^(-1/2);
d(cbrt(x_),x_):= 1/3*x^(-2/3);

d(y_):=d(y,x);
#d(integrate(y_)) := y;