Calculus Examples

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

Step 1

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

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

Step 2

Evaluate $\frac{d}{d x} [\ln (x^{2})]$ .

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Step 2.1

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

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Step 2.1.1

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

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

Step 2.1.2

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

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

Step 2.1.3

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

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

Step 2.2

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

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

Step 2.3

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

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

Step 2.4

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

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

Step 2.5

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

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Step 2.5.1

Factor $x$ out of $2 x$ .

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

Step 2.5.2

Cancel the common factors.

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Step 2.5.2.1

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

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

Step 2.5.2.2

Cancel the common factor.

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

Step 2.5.2.3

Rewrite the expression.

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

Step 3

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

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

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Step 3.1.1

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

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

Step 3.1.2

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

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

Step 3.1.3

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

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

Step 3.2

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

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

Step 3.3

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

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

Step 3.4

Multiply $2$ by $1$ .

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

Step 3.5

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

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