Calculus Examples

$f (x) = \ln (e^{x^{2}} + 1)$

Step 1

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

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Step 1.1

To apply the Chain Rule, set $u_{1}$ as $e^{x^{2}} + 1$ .

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

Step 1.2

The derivative of $\ln (u_{1})$ with respect to $u_{1}$ is $\frac{1}{u_{1}}$ .

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

Step 1.3

Replace all occurrences of $u_{1}$ with $e^{x^{2}} + 1$ .

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

Step 2

By the Sum Rule, the derivative of $e^{x^{2}} + 1$ with respect to $x$ is $\frac{d}{d x} [e^{x^{2}}] + \frac{d}{d x} [1]$ .

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

Step 3

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 $g (x) = x^{2}$ .

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Step 3.1

To apply the Chain Rule, set $u_{2}$ as $x^{2}$ .

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

Step 3.2

Differentiate using the Exponential Rule which states that $\frac{d}{d u_{2}} [a^{u_{2}}]$ is $a^{u_{2}} \ln (a)$ where $a$ = $e$ .

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

Step 3.3

Replace all occurrences of $u_{2}$ with $x^{2}$ .

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

Step 4

Differentiate.

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Step 4.1

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

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

Step 4.2

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

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

Step 4.3

Combine fractions.

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Step 4.3.1

Add $e^{x^{2}} (2 x)$ and $0$ .

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

Step 4.3.2

Combine $e^{x^{2}}$ and $\frac{1}{e^{x^{2}} + 1}$ .

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

Step 4.3.3

Combine $2$ and $\frac{e^{x^{2}}}{e^{x^{2}} + 1}$ .

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

Step 4.3.4

Combine $\frac{2 e^{x^{2}}}{e^{x^{2}} + 1}$ and $x$ .

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

Step 5

Simplify.

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Step 5.1

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

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

Step 5.2

Reorder terms.

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

Step 5.3

Reorder factors in $\frac{2 e^{x^{2}} x}{e^{x^{2}} + 1}$ .

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