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

Aug 10, 2014

$f ' \left(x\right) = {e}^{x} \left(\left(4 {x}^{3} + 1\right) \tan \left(x\right) + \left({x}^{4} + x\right) \tan \left(x\right) + \left({x}^{4} + x\right) {\sec}^{2} x\right)$

Solution

f(x)=(x^4+x)⋅e^x⋅tan(x)

For problems, having more than two functions, like

$f \left(x\right) = u \left(x\right) \cdot v \left(x\right) \cdot w \left(x\right)$

then, differentiating both sides with respect to $x$ using Product Rule, we get

$f ' \left(x\right) = u ' \left(x\right) \cdot v \left(x\right) \cdot w \left(x\right) + u \left(x\right) \cdot v ' \left(x\right) \cdot w \left(x\right) + u \left(x\right) \cdot v \left(x\right) \cdot w ' \left(x\right)$

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)