Calculus Examples

$y = \ln (e^{x} + x e^{x})$

Step 1

Differentiate both sides of the equation.

$\frac{d}{d x} (y) = \frac{d}{d x} (\ln (e^{x} + x e^{x}))$

Step 2

The derivative of $y$ with respect to $x$ is $y'$ .

$y'$

Step 3

Differentiate the right side of the equation.

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Step 3.1

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 $g (x) = e^{x} + x e^{x}$ .

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Step 3.1.1

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

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

Step 3.1.2

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

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

Step 3.1.3

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

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

Step 3.2

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

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

Step 3.3

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

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

Step 3.4

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

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

Step 3.5

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

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

Step 3.6

Differentiate using the Power Rule.

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Step 3.6.1

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

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

Step 3.6.2

Simplify by adding terms.

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Step 3.6.2.1

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

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

Step 3.6.2.2

Add $e^{x}$ and $e^{x}$ .

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

Step 3.7

Simplify.

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Step 3.7.1

Reorder the factors of $\frac{1}{e^{x} + x e^{x}} (x e^{x} + 2 e^{x})$ .

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

Step 3.7.2

Factor $e^{x}$ out of $e^{x} + x e^{x}$ .

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Step 3.7.2.1

Multiply by $1$ .

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

Step 3.7.2.2

Factor $e^{x}$ out of $x e^{x}$ .

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

Step 3.7.2.3

Factor $e^{x}$ out of $e^{x} \cdot 1 + e^{x} x$ .

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

Step 3.7.3

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

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

Step 3.7.4

Factor $e^{x}$ out of $x e^{x} + 2 e^{x}$ .

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Step 3.7.4.1

Factor $e^{x}$ out of $x e^{x}$ .

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

Step 3.7.4.2

Factor $e^{x}$ out of $2 e^{x}$ .

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

Step 3.7.4.3

Factor $e^{x}$ out of $e^{x} x + e^{x} \cdot 2$ .

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

Step 3.7.5

Cancel the common factor of $e^{x}$ .

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Step 3.7.5.1

Cancel the common factor.

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

Step 3.7.5.2

Rewrite the expression.

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

Step 4

Reform the equation by setting the left side equal to the right side.

$y' = \frac{x + 2}{1 + x}$

Step 5

Replace $y'$ with $\frac{d y}{d x}$ .

$\frac{d y}{d x} = \frac{x + 2}{1 + x}$