# How do you differentiate f(x)= (x^2+1)(5x^4+3^2+2x)  using the product rule?

May 7, 2018

color(brown)(f'(x) = 30x^5 + 20x^3 + 6x^2 + 18x + 2

#### Explanation:

$f \left(x\right) = \left({x}^{2} + 1\right) \cdot \left(5 {x}^{4} + 2 x + {3}^{2}\right)$

Let " "u = ((x^2 + 1), v = (5x^4 + 2x + 9)

$\frac{\mathrm{du}}{\mathrm{dx}} = \left(\frac{d}{\mathrm{dx}}\right) \left({x}^{2} + 1\right) = 2 x$

$\frac{\mathrm{dv}}{\mathrm{dx}} = \frac{d}{\mathrm{dx}} \left(5 {x}^{4} + 2 x + 9\right) = 20 {x}^{3} + 2$

$f ' \left(x\right) = v \frac{\mathrm{du}}{\mathrm{dx}} + u \frac{\mathrm{dv}}{\mathrm{dx}}$

$f ' \left(x\right) = \left(5 {x}^{4} + 2 x + 9\right) \cdot 2 x + \left({x}^{2} + 1\right) \cdot \left(20 {x}^{3} + 2\right)$

$\implies 10 {x}^{5} + 4 {x}^{2} + 18 x + 20 {x}^{5} + 2 {x}^{2} + 20 {x}^{3} + 2$

color(brown)(=> 30x^5 + 20x^3 + 6x^2 + 18x + 2