# What is the derivative of (x^3 + 2)^2(x^5 + 4)^4?

Jun 29, 2018

${f}^{'} \left(x\right) = 2 {x}^{2} \left({x}^{3} + 2\right) {\left({x}^{5} + 4\right)}^{3} \left(13 {x}^{5} + 20 {x}^{2} + 12\right)$

#### Explanation:

$f \left(x\right) = {\left({x}^{3} + 2\right)}^{2} {\left({x}^{5} + 4\right)}^{4}$

Product rule : $\frac{d}{d} x \left(f g\right) = f {g}^{'} + g {f}^{'}$

${f}^{'} \left(x\right) = {\left({x}^{3} + 2\right)}^{2} \cdot 4 {\left({x}^{5} + 4\right)}^{3} \cdot \left(5 {x}^{4}\right) + {\left({x}^{5} + 4\right)}^{4} \cdot 2 \left({x}^{3} + 2\right) \cdot \left(3 {x}^{2}\right)$

${f}^{'} \left(x\right) = 2 {x}^{2} \left({x}^{3} + 2\right) {\left({x}^{5} + 4\right)}^{3} \left\{\left(10 {x}^{5} + 20 {x}^{2}\right) + \left(3 {x}^{5} + 12\right)\right\}$

${f}^{'} \left(x\right) = 2 {x}^{2} \left({x}^{3} + 2\right) {\left({x}^{5} + 4\right)}^{3} \left(13 {x}^{5} + 20 {x}^{2} + 12\right)$ [Ans]