# How do you differentiate f(x)=sqrt(e^(5x^2+x+3)  using the chain rule?

Jul 10, 2016

This will require the application of the chain rule--twice.

First, for ${e}^{5 {x}^{2} + x + 3}$

Let $y = {e}^{u}$, and $u = 5 {x}^{2} + x + 3$

$\frac{\mathrm{dy}}{\mathrm{dx}} = \frac{\mathrm{dy}}{\mathrm{du}} \times \frac{\mathrm{du}}{\mathrm{dx}}$

The derivative of ${e}^{u}$ is ${e}^{u}$. The derivative of $5 {x}^{2} + x + 3$ is $10 x + 1$.

Hence,

$\frac{\mathrm{dy}}{\mathrm{dx}} = {e}^{u} \times 10 x + 1$

$\frac{\mathrm{dy}}{\mathrm{dx}} = \left(10 x + 1\right) {e}^{5 {x}^{2} + x + 3}$

Now for the second application of the chain rule.

Let $y = \sqrt{u} = {u}^{\frac{1}{2}}$ and $u = {e}^{5 {x}^{2} + x + 3}$

We already know the derivative of $u$. The derivative of $y$, by the power rule, is $\frac{1}{2} {u}^{- \frac{1}{2}} = \frac{1}{2 {u}^{\frac{1}{2}}}$.

Hence,

$\frac{\mathrm{dy}}{\mathrm{dx}} = \frac{1}{2 {u}^{\frac{1}{2}}} \times \left(10 x + 1\right) {e}^{5 {x}^{2} + x + 3}$

dy/dx = ((10x + 1)e^(5x^2 + x + 3))/(2sqrt(e^(5x^2 + x + 3))

f'(x) = ((10x + 1)e^(5x^2 + x + 3))/(2sqrt(e^(5x^2 + x + 3))

f'(x) = ((10x + 1)e^(5x^2 + x + 3))/(2(e^(5x^2 + x + 3))^(1/2)

By the quotient rule of exponents: ${a}^{n} / {a}^{m} = {a}^{n - m}$:

$f ' \left(x\right) = \frac{\left(10 x + 1\right) \sqrt{{e}^{5 {x}^{2} + x + 3}}}{2}$

Hopefully this helps!