# How do you apply the product rule repeatedly to find the derivative of #f(x) = (x^4 +x)*e^x*tan(x)# ?

##### 1 Answer

Aug 10, 2014

#f'(x)=e^x((4x^3+1)tan(x)+(x^4+x)tan(x)+(x^4+x)sec^2x)#

Solution

#f(x)=(x^4+x)⋅e^x⋅tan(x)# For problems, having more than two functions, like

#f(x)=u(x)*v(x)*w(x)# then, differentiating both sides with respect to

#x# using Product Rule, we get

#f'(x)=u'(x)*v(x)*w(x)+u(x)*v'(x)*w(x)+u(x)*v(x)*w'(x)# similarly, following the same pattern for the given problem and differentiating with respect to

#x# ,

#f'(x)=(4x^3+1)⋅e^x⋅tan(x)+(x^4+x)⋅e^x⋅tan(x)+(x^4+x)⋅e^x⋅sec^2x# simplifying further,

#f'(x)=e^x*((4x^3+1)⋅tan(x)+(x^4+x)⋅tan(x)+(x^4+x)⋅sec^2x)#