Calculus Examples

$y = \cosh (\ln (4 x^{3}))$

Step 1

Differentiate using the chain rule, which states that $\frac{d}{d x} [f (g (x))]$ is $f' (g (x)) g' (x)$ where $f (x) = \cosh (x)$ and $g (x) = \ln (4 x^{3})$ .

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Step 1.1

To apply the Chain Rule, set $u_{1}$ as $\ln (4 x^{3})$ .

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

Step 1.2

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

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

Step 1.3

Replace all occurrences of $u_{1}$ with $\ln (4 x^{3})$ .

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

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 $g (x) = 4 x^{3}$ .

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Step 2.1

To apply the Chain Rule, set $u_{2}$ as $4 x^{3}$ .

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

Step 2.2

The derivative of $\ln (u_{2})$ with respect to $u_{2}$ is $\frac{1}{u_{2}}$ .

$\sinh (\ln (4 x^{3})) (\frac{1}{u_{2}} \frac{d}{d x} [4 x^{3}])$

Step 2.3

Replace all occurrences of $u_{2}$ with $4 x^{3}$ .

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

Step 3

Differentiate.

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Step 3.1

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

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

Step 3.2

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

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

Step 3.3

Simplify terms.

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Step 3.3.1

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

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

Step 3.3.2

Cancel the common factor of $4$ .

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Step 3.3.2.1

Cancel the common factor.

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

Step 3.3.2.2

Rewrite the expression.

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

Step 3.4

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

$\frac{\sinh (\ln (4 x^{3}))}{x^{3}} (3 x^{2})$

Step 3.5

Simplify terms.

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Step 3.5.1

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

$\frac{3 \sinh (\ln (4 x^{3}))}{x^{3}} x^{2}$

Step 3.5.2

Combine $\frac{3 \sinh (\ln (4 x^{3}))}{x^{3}}$ and $x^{2}$ .

$\frac{3 \sinh (\ln (4 x^{3})) x^{2}}{x^{3}}$

Step 3.5.3

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

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Step 3.5.3.1

Factor $x^{2}$ out of $3 \sinh (\ln (4 x^{3})) x^{2}$ .

$\frac{x^{2} (3 \sinh (\ln (4 x^{3})))}{x^{3}}$

Step 3.5.3.2

Cancel the common factors.

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Step 3.5.3.2.1

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

$\frac{x^{2} (3 \sinh (\ln (4 x^{3})))}{x^{2} x}$

Step 3.5.3.2.2

Cancel the common factor.

$\frac{x^{2} (3 \sinh (\ln (4 x^{3})))}{x^{2} x}$

Step 3.5.3.2.3

Rewrite the expression.

$\frac{3 \sinh (\ln (4 x^{3}))}{x}$