Calculus Examples

$f (x) = \cos (3 x) \ln (x^{4})$

Step 1

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 $f (x) = \cos (3 x)$ and $g (x) = \ln (x^{4})$ .

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

Step 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 $g (x) = x^{4}$ .

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Step 2.1

To apply the Chain Rule, set $u_{1}$ as $x^{4}$ .

$\cos (3 x) (\frac{d}{d u_{1}} [\ln (u_{1})] \frac{d}{d x} [x^{4}]) + \ln (x^{4}) \frac{d}{d x} [\cos (3 x)]$

Step 2.2

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

$\cos (3 x) (\frac{1}{u_{1}} \frac{d}{d x} [x^{4}]) + \ln (x^{4}) \frac{d}{d x} [\cos (3 x)]$

Step 2.3

Replace all occurrences of $u_{1}$ with $x^{4}$ .

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

Step 3

Differentiate using the Power Rule.

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Step 3.1

Combine $\frac{1}{x^{4}}$ and $\cos (3 x)$ .

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

Step 3.2

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 4$ .

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

Step 3.3

Simplify terms.

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Step 3.3.1

Combine $4$ and $\frac{\cos (3 x)}{x^{4}}$ .

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

Step 3.3.2

Combine $\frac{4 \cos (3 x)}{x^{4}}$ and $x^{3}$ .

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

Step 3.3.3

Cancel the common factor of $x^{3}$ and $x^{4}$ .

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Step 3.3.3.1

Factor $x^{3}$ out of $4 \cos (3 x) x^{3}$ .

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

Step 3.3.3.2

Cancel the common factors.

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Step 3.3.3.2.1

Factor $x^{3}$ out of $x^{4}$ .

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

Step 3.3.3.2.2

Cancel the common factor.

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

Step 3.3.3.2.3

Rewrite the expression.

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

Step 4

Differentiate using the chain rule, which states that $\frac{d}{d x} [f (g (x))]$ is $f' (g (x)) g' (x)$ where $f (x) = \cos (x)$ and $g (x) = 3 x$ .

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Step 4.1

To apply the Chain Rule, set $u_{2}$ as $3 x$ .

$\frac{4 \cos (3 x)}{x} + \ln (x^{4}) (\frac{d}{d u_{2}} [\cos (u_{2})] \frac{d}{d x} [3 x])$

Step 4.2

The derivative of $\cos (u_{2})$ with respect to $u_{2}$ is $- \sin (u_{2})$ .

$\frac{4 \cos (3 x)}{x} + \ln (x^{4}) (- \sin (u_{2}) \frac{d}{d x} [3 x])$

Step 4.3

Replace all occurrences of $u_{2}$ with $3 x$ .

$\frac{4 \cos (3 x)}{x} + \ln (x^{4}) (- \sin (3 x) \frac{d}{d x} [3 x])$

Step 5

Differentiate.

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Step 5.1

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

$\frac{4 \cos (3 x)}{x} + \ln (x^{4}) (- \sin (3 x) (3 \frac{d}{d x} [x]))$

Step 5.2

Multiply $3$ by $- 1$ .

$\frac{4 \cos (3 x)}{x} + \ln (x^{4}) (- 3 \sin (3 x) \frac{d}{d x} [x])$

Step 5.3

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

$\frac{4 \cos (3 x)}{x} + \ln (x^{4}) (- 3 \sin (3 x) \cdot 1)$

Step 5.4

Simplify the expression.

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Step 5.4.1

Multiply $- 3$ by $1$ .

$\frac{4 \cos (3 x)}{x} + \ln (x^{4}) (- 3 \sin (3 x))$

Step 5.4.2

Reorder terms.

$- 3 \sin (3 x) \ln (x^{4}) + \frac{4 \cos (3 x)}{x}$