Calculus Examples

$f (x) = x^{5} e^{2 x}$

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 $f (x) = x^{5}$ and $g (x) = e^{2 x}$ .

$x^{5} \frac{d}{d x} [e^{2 x}] + e^{2 x} \frac{d}{d x} [x^{5}]$

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 $g (x) = 2 x$ .

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Step 2.1

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

$x^{5} (\frac{d}{d u} [e^{u}] \frac{d}{d x} [2 x]) + e^{2 x} \frac{d}{d x} [x^{5}]$

Step 2.2

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

$x^{5} (e^{u} \frac{d}{d x} [2 x]) + e^{2 x} \frac{d}{d x} [x^{5}]$

Step 2.3

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

$x^{5} (e^{2 x} \frac{d}{d x} [2 x]) + e^{2 x} \frac{d}{d x} [x^{5}]$

Step 3

Differentiate.

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Step 3.1

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

$x^{5} (e^{2 x} (2 \frac{d}{d x} [x])) + e^{2 x} \frac{d}{d x} [x^{5}]$

Step 3.2

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

$x^{5} (e^{2 x} (2 \cdot 1)) + e^{2 x} \frac{d}{d x} [x^{5}]$

Step 3.3

Simplify the expression.

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Step 3.3.1

Multiply $2$ by $1$ .

$x^{5} (e^{2 x} \cdot 2) + e^{2 x} \frac{d}{d x} [x^{5}]$

Step 3.3.2

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

$x^{5} (2 \cdot e^{2 x}) + e^{2 x} \frac{d}{d x} [x^{5}]$

Step 3.4

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 5$ .

$x^{5} (2 e^{2 x}) + e^{2 x} (5 x^{4})$

Step 4

Simplify.

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Step 4.1

Reorder terms.

$2 e^{2 x} x^{5} + 5 e^{2 x} x^{4}$

Step 4.2

Reorder factors in $2 e^{2 x} x^{5} + 5 e^{2 x} x^{4}$ .

$2 x^{5} e^{2 x} + 5 x^{4} e^{2 x}$