Find the Derivative - d/dx xe^(2x)

$x e^{2 x}$

Step 1

Differentiate using the Product Rule which states that $\frac{d}{d x} [f (x) g (x)]$ is $f (x) \frac{d}{d x} [g (x)] + g (x) \frac{d}{d x} [f (x)]$ where $f (x) = x$ and $g (x) = e^{2 x}$ .

$x \frac{d}{d x} [e^{2 x}] + e^{2 x} \frac{d}{d x} [x]$

Step 2

Differentiate using the chain rule, which states that $\frac{d}{d x} [f (g (x))]$ is $f' (g (x)) g' (x)$ where $f (x) = e^{x}$ and $g (x) = 2 x$ .

To apply the Chain Rule, set $u$ as $2 x$ .

$x (\frac{d}{d u} [e^{u}] \frac{d}{d x} [2 x]) + e^{2 x} \frac{d}{d x} [x]$

Differentiate using the Exponential Rule which states that $\frac{d}{d u} [a^{u}]$ is $a^{u} \ln (a)$ where $a$ = $e$ .

$x (e^{u} \frac{d}{d x} [2 x]) + e^{2 x} \frac{d}{d x} [x]$

Replace all occurrences of $u$ with $2 x$ .

$x (e^{2 x} \frac{d}{d x} [2 x]) + e^{2 x} \frac{d}{d x} [x]$

Step 3

Differentiate.

Since $2$ is constant with respect to $x$ , the derivative of $2 x$ with respect to $x$ is $2 \frac{d}{d x} [x]$ .

$x (e^{2 x} (2 \frac{d}{d x} [x])) + e^{2 x} \frac{d}{d x} [x]$

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 1$ .

$x (e^{2 x} (2 \cdot 1)) + e^{2 x} \frac{d}{d x} [x]$

Simplify the expression.

Multiply $2$ by $1$ .

$x (e^{2 x} \cdot 2) + e^{2 x} \frac{d}{d x} [x]$

Move $2$ to the left of $e^{2 x}$ .

$x (2 \cdot e^{2 x}) + e^{2 x} \frac{d}{d x} [x]$

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 1$ .

$x (2 e^{2 x}) + e^{2 x} \cdot 1$

Multiply $e^{2 x}$ by $1$ .

$x (2 e^{2 x}) + e^{2 x}$

Step 4

Simplify.

Reorder terms.

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

Reorder factors in $2 e^{2 x} x + e^{2 x}$ .

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

Find the Derivative - d/dx xe^(2x)

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