Calculus Examples

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

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

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Step 1.1

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

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

Step 1.2

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

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

Step 1.3

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

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

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

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

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 $g (x) = 2 x$ .

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Step 3.1

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

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

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 $a$ = $e$ .

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

Step 3.3

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

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

Step 4

Differentiate.

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Step 4.1

Since $2$ is constant with respect to $x$ , the derivative of $2 x$ with respect to $x$ is $2 \frac{d}{d x} [x]$ .

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

Step 4.2

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

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

Step 4.3

Simplify the expression.

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Step 4.3.1

Multiply $2$ by $1$ .

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

Step 4.3.2

Move $2$ to the left of $e^{2 x}$ .

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

Step 4.4

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

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

Step 5

Simplify.

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Step 5.1

Apply the distributive property.

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

Step 5.2

Combine terms.

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Step 5.2.1

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

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

Step 5.2.2

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

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

Step 5.2.3

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

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

Step 5.2.4

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

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Step 5.2.4.1

Cancel the common factor.

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

Step 5.2.4.2

Rewrite the expression.

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

Step 5.2.5

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

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Step 5.2.5.1

Cancel the common factor.

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

Step 5.2.5.2

Divide $2$ by $1$ .

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

Step 5.2.6

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

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

Step 5.2.7

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

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

Step 5.2.8

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

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

Step 5.2.9

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

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Step 5.2.9.1

Cancel the common factor.

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

Step 5.2.9.2

Rewrite the expression.

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

Step 5.2.10

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

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Step 5.2.10.1

Factor $x$ out of $2 x$ .

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

Step 5.2.10.2

Cancel the common factors.

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Step 5.2.10.2.1

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

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

Step 5.2.10.2.2

Cancel the common factor.

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

Step 5.2.10.2.3

Rewrite the expression.

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

Step 5.3

Reorder terms.

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