Calculus Examples

$f (x) = 4 \ln (x) + \ln (x^{6}) + 8 e^{x}$

Step 1

By the Sum Rule, the derivative of $4 \ln (x) + \ln (x^{6}) + 8 e^{x}$ with respect to $x$ is $\frac{d}{d x} [4 \ln (x)] + \frac{d}{d x} [\ln (x^{6})] + \frac{d}{d x} [8 e^{x}]$ .

$\frac{d}{d x} [4 \ln (x)] + \frac{d}{d x} [\ln (x^{6})] + \frac{d}{d x} [8 e^{x}]$

Step 2

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

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Step 2.1

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

$4 \frac{d}{d x} [\ln (x)] + \frac{d}{d x} [\ln (x^{6})] + \frac{d}{d x} [8 e^{x}]$

Step 2.2

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

$4 \frac{1}{x} + \frac{d}{d x} [\ln (x^{6})] + \frac{d}{d x} [8 e^{x}]$

Step 2.3

Combine $4$ and $\frac{1}{x}$ .

$\frac{4}{x} + \frac{d}{d x} [\ln (x^{6})] + \frac{d}{d x} [8 e^{x}]$

Step 3

Evaluate $\frac{d}{d x} [\ln (x^{6})]$ .

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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 $g (x) = x^{6}$ .

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Step 3.1.1

To apply the Chain Rule, set $u$ as $x^{6}$ .

$\frac{4}{x} + \frac{d}{d u} [\ln (u)] \frac{d}{d x} [x^{6}] + \frac{d}{d x} [8 e^{x}]$

Step 3.1.2

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

$\frac{4}{x} + \frac{1}{u} \frac{d}{d x} [x^{6}] + \frac{d}{d x} [8 e^{x}]$

Step 3.1.3

Replace all occurrences of $u$ with $x^{6}$ .

$\frac{4}{x} + \frac{1}{x^{6}} \frac{d}{d x} [x^{6}] + \frac{d}{d x} [8 e^{x}]$

Step 3.2

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 6$ .

$\frac{4}{x} + \frac{1}{x^{6}} (6 x^{5}) + \frac{d}{d x} [8 e^{x}]$

Step 3.3

Combine $6$ and $\frac{1}{x^{6}}$ .

$\frac{4}{x} + \frac{6}{x^{6}} x^{5} + \frac{d}{d x} [8 e^{x}]$

Step 3.4

Combine $\frac{6}{x^{6}}$ and $x^{5}$ .

$\frac{4}{x} + \frac{6 x^{5}}{x^{6}} + \frac{d}{d x} [8 e^{x}]$

Step 3.5

Cancel the common factor of $x^{5}$ and $x^{6}$ .

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Step 3.5.1

Factor $x^{5}$ out of $6 x^{5}$ .

$\frac{4}{x} + \frac{x^{5} \cdot 6}{x^{6}} + \frac{d}{d x} [8 e^{x}]$

Step 3.5.2

Cancel the common factors.

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Step 3.5.2.1

Factor $x^{5}$ out of $x^{6}$ .

$\frac{4}{x} + \frac{x^{5} \cdot 6}{x^{5} x} + \frac{d}{d x} [8 e^{x}]$

Step 3.5.2.2

Cancel the common factor.

$\frac{4}{x} + \frac{x^{5} \cdot 6}{x^{5} x} + \frac{d}{d x} [8 e^{x}]$

Step 3.5.2.3

Rewrite the expression.

$\frac{4}{x} + \frac{6}{x} + \frac{d}{d x} [8 e^{x}]$

Step 4

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

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Step 4.1

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

$\frac{4}{x} + \frac{6}{x} + 8 \frac{d}{d x} [e^{x}]$

Step 4.2

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

$\frac{4}{x} + \frac{6}{x} + 8 e^{x}$

Step 5

Combine terms.

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Step 5.1

Combine the numerators over the common denominator.

$\frac{4 + 6}{x} + 8 e^{x}$

Step 5.2

Add $4$ and $6$ .

$\frac{10}{x} + 8 e^{x}$