Calculus Examples

$y = \ln (x^{2} e^{x})$

Step 1

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

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Step 1.1

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

$\frac{d}{d u} [\ln (u)] \frac{d}{d x} [x^{2} e^{x}]$

Step 1.2

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

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

Step 1.3

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

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

Step 2

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 $f (x) = x^{2}$ and $g (x) = e^{x}$ .

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

Step 3

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

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

Step 4

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

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

Step 5

Simplify.

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Step 5.1

Apply the distributive property.

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

Step 5.2

Combine terms.

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Step 5.2.1

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

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

Step 5.2.2

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

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

Step 5.2.3

Cancel the common factor of $x^{2}$ .

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Step 5.2.3.1

Cancel the common factor.

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

Step 5.2.3.2

Rewrite the expression.

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

Step 5.2.4

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

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Step 5.2.4.1

Cancel the common factor.

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

Step 5.2.4.2

Rewrite the expression.

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

Step 5.2.5

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

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

Step 5.2.6

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

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

Step 5.2.7

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

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

Step 5.2.8

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

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Step 5.2.8.1

Cancel the common factor.

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

Step 5.2.8.2

Rewrite the expression.

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

Step 5.2.9

Cancel the common factor of $x$ and $x^{2}$ .

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Step 5.2.9.1

Factor $x$ out of $2 x$ .

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

Step 5.2.9.2

Cancel the common factors.

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Step 5.2.9.2.1

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

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

Step 5.2.9.2.2

Cancel the common factor.

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

Step 5.2.9.2.3

Rewrite the expression.

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

Step 5.3

Reorder terms.

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