# If f(x)= 2 x^2 + x  and g(x) = 2e^x + 1 , how do you differentiate f(g(x))  using the chain rule?

Jan 3, 2018

$16 {e}^{2 x} + 10 {e}^{x}$

#### Explanation:

f(g(x)

$= f \left(2 {e}^{x} + 1\right) = 2 {\left(2 {e}^{x} + 1\right)}^{2} + 2 {e}^{x} + 1$

$\textcolor{w h i t e}{\times \times \times \times x} = 2 \left(4 {e}^{2 x} + 4 {e}^{x} + 1\right) + 2 {e}^{x} + 1$

$\textcolor{w h i t e}{\times \times \times \times x} = 8 {e}^{2 x} + 8 {e}^{x} + 2 + 2 {e}^{x} + 1$

$\textcolor{w h i t e}{\times \times \times \times x} = 8 {e}^{2 x} + 10 {e}^{x} + 3$

$\Rightarrow \frac{d}{\mathrm{dx}} \left(8 {e}^{2 x} + 10 {e}^{x} + 3\right)$

$= 8 {e}^{2 x} \times \frac{d}{\mathrm{dx}} \left(2 x\right) + 10 {e}^{x} \leftarrow \textcolor{b l u e}{\text{chain rule}}$

$= 16 {e}^{2 x} + 10 {e}^{x}$

Jan 3, 2018

See below.

#### Explanation:

$f \left(x\right) = 2 {x}^{2} + x$ , $g \left(x\right) = 2 {e}^{x} + 1$

$\therefore$

$f \left(g \left(x\right)\right) = 2 {\left(2 {e}^{x} + 1\right)}^{2} + 2 {e}^{x} + 1 = 8 {e}^{2 x} + 10 {e}^{x} + 3$

Using the chain rule:

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

Let $u = {e}^{x}$

$8 {u}^{2} + 10 u + 3$

$\frac{\mathrm{dy}}{\mathrm{dx}} \left(8 {u}^{2}\right) = \frac{\mathrm{dy}}{\mathrm{du}} \left(8 {u}^{2}\right) \cdot \frac{\mathrm{du}}{\mathrm{dx}} \left(u\right) = 16 u \cdot {e}^{x} = 16 {e}^{2 x}$

$\frac{\mathrm{dy}}{\mathrm{dx}} \left(10 u\right) = \frac{\mathrm{dy}}{\mathrm{du}} \left(10 u\right) \cdot \frac{\mathrm{du}}{\mathrm{dx}} \left(u\right) = 10 \cdot {e}^{x} = 10 {e}^{x}$

$\therefore$

$\frac{\mathrm{dy}}{\mathrm{dx}} \left(8 {x}^{2 x} + 10 {e}^{x} + 3\right) = 16 {e}^{2 x} + 10 {e}^{x}$