Calculus Examples

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

Step 1

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

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Step 1.1

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

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

Step 1.2

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

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

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 $g (x) = 1 + x$ .

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Step 2.1

To apply the Chain Rule, set $u$ as $1 + x$ .

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

Step 2.2

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

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

Step 2.3

Replace all occurrences of $u$ with $1 + x$ .

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

Step 3

Differentiate.

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Step 3.1

By the Sum Rule, the derivative of $1 + x$ with respect to $x$ is $\frac{d}{d x} [1] + \frac{d}{d x} [x]$ .

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

Step 3.2

Since $1$ is constant with respect to $x$ , the derivative of $1$ with respect to $x$ is $0$ .

$e^{1 + x} (0 + \frac{d}{d x} [x])$

Step 3.3

Add $0$ and $\frac{d}{d x} [x]$ .

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

Step 3.4

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

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

Step 3.5

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

$e^{1 + x}$