Question #5240a Calculus Differentiating Trigonometric Functions Differentiating Inverse Trigonometric Functions 1 Answer Mark Mar 27, 2017 csc(xln(x))= -csc(xln(x))cot(xln(x) )*(ln(x) + 1) Explanation: csc(u(x))' = -csc(u(x))cot(u(x)) ln(x)' = 1/x Formula for multiplication: (f * g)(a) = f'(a)g(a) + g'(a)f(a) (xln(x))'= d/dx (x)*ln(x) + d/dx ln(x)*x = (ln(x) + 1) csc(xln(x)) = -csc(xln(x))cot(xln(x) )* d/dx (xln(x)) csc(xln(x))= -csc(xln(x))cot(xln(x) )*(ln(x) + 1) --------------------- to prove it: int (-csc(xln(x))cot(xln(x) )*(ln(x) + 1))dx u = xlnx du= (ln(x) + 1) int (-csc(u)cot(u ) = csc(u) int(-csc(u)cot(u )*du) = (csc(u)) = csc(xln(x)) Answer link Related questions What is the derivative of f(x)=sin^-1(x) ? What is the derivative of f(x)=cos^-1(x) ? What is the derivative of f(x)=tan^-1(x) ? What is the derivative of f(x)=sec^-1(x) ? What is the derivative of f(x)=csc^-1(x) ? What is the derivative of f(x)=cot^-1(x) ? What is the derivative of f(x)=(cos^-1(x))/x ? What is the derivative of f(x)=tan^-1(e^x) ? What is the derivative of f(x)=cos^-1(x^3) ? What is the derivative of f(x)=ln(sin^-1(x)) ? See all questions in Differentiating Inverse Trigonometric Functions Impact of this question 1556 views around the world You can reuse this answer Creative Commons License