# Differentiate the given function: f(x)=-3(5x^3-2x+5)(\sqrt x +2x)

## Question:

Differentiate the given function: {eq}f(x)=-3(5x^3-2x+5)(\sqrt x +2x) {/eq}

## The Quotient Rule of Differentiation:

We are given a function f(x) defined as the product between two functions g(x) and h(x)

{eq}\displaystyle f(x) = g(x) h(x) {/eq}

The derivative of the function f is computed by means of the product rule of differentiation

{eq}\displaystyle f'(x) = g'(x) h(x) + g(x) h'(x). {/eq}

The derivative of the function

{eq}\displaystyle f(x)=-3(5x^3-2x+5)(\sqrt x +2x) {/eq}

is computed by means of the product rule of differentiation

{eq}\displaystyle (g(x)h(x))' = g'(x) h(x) + g(x) h'(x). {/eq}

Therefore we get:

{eq}\displaystyle f'(x)= -3\frac{d}{dx}\left(\left(5x^3-2x+5\right)\sqrt{x+2x}\right) \\ \displaystyle = 3\left(\frac{d}{dx}\left(5x^3-2x+5\right)\sqrt{x+2x}+\frac{d}{dx}\left(\sqrt{x+2x}\right)\left(5x^3-2x+5\right)\right) \\ \displaystyle = -3\left(\left(15x^2-2\right)\sqrt{x+2x}+\frac{\sqrt{3}}{2\sqrt{x}}\left(5x^3-2x+5\right)\right). {/eq}