Calculus Examples

$f (x) = 8 e^{x} + 4 \ln (x^{3})$

Step 1

By the Sum Rule, the derivative of $8 e^{x} + 4 \ln (x^{3})$ with respect to $x$ is $\frac{d}{d x} [8 e^{x}] + \frac{d}{d x} [4 \ln (x^{3})]$ .

$\frac{d}{d x} [8 e^{x}] + \frac{d}{d x} [4 \ln (x^{3})]$

Step 2

Evaluate $\frac{d}{d x} [8 e^{x}]$ .

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Step 2.1

Since $8$ is constant with respect to $x$ , the derivative of $8 e^{x}$ with respect to $x$ is $8 \frac{d}{d x} [e^{x}]$ .

$8 \frac{d}{d x} [e^{x}] + \frac{d}{d x} [4 \ln (x^{3})]$

Step 2.2

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

$8 e^{x} + \frac{d}{d x} [4 \ln (x^{3})]$

Step 3

Evaluate $\frac{d}{d x} [4 \ln (x^{3})]$ .

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Step 3.1

Since $4$ is constant with respect to $x$ , the derivative of $4 \ln (x^{3})$ with respect to $x$ is $4 \frac{d}{d x} [\ln (x^{3})]$ .

$8 e^{x} + 4 \frac{d}{d x} [\ln (x^{3})]$

Step 3.2

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 $g (x) = x^{3}$ .

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Step 3.2.1

To apply the Chain Rule, set $u$ as $x^{3}$ .

$8 e^{x} + 4 (\frac{d}{d u} [\ln (u)] \frac{d}{d x} [x^{3}])$

Step 3.2.2

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

$8 e^{x} + 4 (\frac{1}{u} \frac{d}{d x} [x^{3}])$

Step 3.2.3

Replace all occurrences of $u$ with $x^{3}$ .

$8 e^{x} + 4 (\frac{1}{x^{3}} \frac{d}{d x} [x^{3}])$

Step 3.3

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 3$ .

$8 e^{x} + 4 (\frac{1}{x^{3}} (3 x^{2}))$

Step 3.4

Combine $3$ and $\frac{1}{x^{3}}$ .

$8 e^{x} + 4 (\frac{3}{x^{3}} x^{2})$

Step 3.5

Combine $\frac{3}{x^{3}}$ and $x^{2}$ .

$8 e^{x} + 4 \frac{3 x^{2}}{x^{3}}$

Step 3.6

Cancel the common factor of $x^{2}$ and $x^{3}$ .

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Step 3.6.1

Factor $x^{2}$ out of $3 x^{2}$ .

$8 e^{x} + 4 \frac{x^{2} \cdot 3}{x^{3}}$

Step 3.6.2

Cancel the common factors.

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Step 3.6.2.1

Factor $x^{2}$ out of $x^{3}$ .

$8 e^{x} + 4 \frac{x^{2} \cdot 3}{x^{2} x}$

Step 3.6.2.2

Cancel the common factor.

$8 e^{x} + 4 \frac{x^{2} \cdot 3}{x^{2} x}$

Step 3.6.2.3

Rewrite the expression.

$8 e^{x} + 4 \frac{3}{x}$

Step 3.7

Combine $4$ and $\frac{3}{x}$ .

$8 e^{x} + \frac{4 \cdot 3}{x}$

Step 3.8

Multiply $4$ by $3$ .

$8 e^{x} + \frac{12}{x}$

Step 4

Reorder terms.

$\frac{12}{x} + 8 e^{x}$