What is the derivative of #[tan(abs(x))]^(-1) #? Calculus Differentiating Trigonometric Functions Differentiating Inverse Trigonometric Functions 1 Answer Monzur R. May 30, 2017 #d/dx(cot|x|)={(-csc^2x, x>=0),(csc^2x, x< 0):}# Explanation: We have #(tan|x|)^-1=cot|x|# #{(cotx, x>=0),(-cotx, x <0):}# Note that #cot(-x)=cos(-x)/sin(-x)=cosx/-sinx=-cotx# #cotx=cosx/sinx# #d/dx(cotx)=d/dx(cosx/sinx)# #d/dx(cosx/sinx)=(-sin^2x-cos^2x)/sin^2x=-1/sin^2x=-csc^2x# #therefored/dx(-cotx)=csc^2x# #d/dx(cot|x|)={(-csc^2x, x>=0),(csc^2x, x< 0):}# 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 1146 views around the world You can reuse this answer Creative Commons License