Calculus Examples

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

Step 1

Differentiate using the chain rule, which states that $\frac{d}{d x} [f (g (x))]$ is $f' (g (x)) g' (x)$ where $f (x) = x^{2}$ and $g (x) = 1 + e^{x}$ .

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Step 1.1

To apply the Chain Rule, set $u$ as $1 + e^{x}$ .

$\frac{d}{d u} [u^{2}] \frac{d}{d x} [1 + e^{x}]$

Step 1.2

Differentiate using the Power Rule which states that $\frac{d}{d u} [u^{n}]$ is $n u^{n - 1}$ where $n = 2$ .

$2 u \frac{d}{d x} [1 + e^{x}]$

Step 1.3

Replace all occurrences of $u$ with $1 + e^{x}$ .

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

Step 2

Differentiate.

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Step 2.1

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

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

Step 2.2

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

$2 (1 + e^{x}) (0 + \frac{d}{d x} [e^{x}])$

Step 2.3

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

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

Step 3

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

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

Step 4

Simplify.

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Step 4.1

Apply the distributive property.

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

Step 4.2

Apply the distributive property.

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

Step 4.3

Combine terms.

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Step 4.3.1

Multiply $2$ by $1$ .

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

Step 4.3.2

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

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Step 4.3.2.1

Move $e^{x}$ .

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

Step 4.3.2.2

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

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

Step 4.3.2.3

Add $x$ and $x$ .

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