Question

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

Answer 1

`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)`