Calculus Examples

$e^{e x}$

Step 1

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 $g (x) = e x$ .

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Step 1.1

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

$\frac{d}{d u} [e^{u}] \frac{d}{d x} [e x]$

Step 1.2

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

$e^{u} \frac{d}{d x} [e x]$

Step 1.3

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

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

Step 2

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

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

Step 3

Multiply $e^{e x}$ by $e$ by adding the exponents.

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Step 3.1

Move $e$ .

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

Step 3.2

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

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Step 3.2.1

Raise $e$ to the power of $1$ .

$e^{1} e^{e x} \frac{d}{d x} [x]$

Step 3.2.2

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$e^{1 + e x} \frac{d}{d x} [x]$

Step 4

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

$e^{1 + e x} \cdot 1$

Step 5

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

$e^{1 + e x}$