Calculus Examples

$f (x) = 4 {(\ln (x) + 3)}^{4} + {(\ln (x) + 3)}^{3} + 2$

Step 1

By the Sum Rule, the derivative of $4 {(\ln (x) + 3)}^{4} + {(\ln (x) + 3)}^{3} + 2$ with respect to $x$ is $\frac{d}{d x} [4 {(\ln (x) + 3)}^{4}] + \frac{d}{d x} [{(\ln (x) + 3)}^{3}] + \frac{d}{d x} [2]$ .

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

Step 2

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

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Step 2.1

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

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

Step 2.2

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

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Step 2.2.1

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

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

Step 2.2.2

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

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

Step 2.2.3

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

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

Step 2.3

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

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

Step 2.4

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

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

Step 2.5

Since $3$ is constant with respect to $x$ , the derivative of $3$ with respect to $x$ is $0$ .

$4 (4 {(\ln (x) + 3)}^{3} (\frac{1}{x} + 0)) + \frac{d}{d x} [{(\ln (x) + 3)}^{3}] + \frac{d}{d x} [2]$

Step 2.6

Add $\frac{1}{x}$ and $0$ .

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

Step 2.7

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

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

Step 2.8

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

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

Step 2.9

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

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

Step 2.10

Multiply $4$ by $4$ .

$\frac{16 {(\ln (x) + 3)}^{3}}{x} + \frac{d}{d x} [{(\ln (x) + 3)}^{3}] + \frac{d}{d x} [2]$

Step 3

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

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

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Step 3.1.1

To apply the Chain Rule, set $u_{2}$ as $\ln (x) + 3$ .

$\frac{16 {(\ln (x) + 3)}^{3}}{x} + \frac{d}{d u_{2}} [{u_{2}}^{3}] \frac{d}{d x} [\ln (x) + 3] + \frac{d}{d x} [2]$

Step 3.1.2

Differentiate using the Power Rule which states that $\frac{d}{d u_{2}} [{u_{2}}^{n}]$ is $n {u_{2}}^{n - 1}$ where $n = 3$ .

$\frac{16 {(\ln (x) + 3)}^{3}}{x} + 3 {u_{2}}^{2} \frac{d}{d x} [\ln (x) + 3] + \frac{d}{d x} [2]$

Step 3.1.3

Replace all occurrences of $u_{2}$ with $\ln (x) + 3$ .

$\frac{16 {(\ln (x) + 3)}^{3}}{x} + 3 {(\ln (x) + 3)}^{2} \frac{d}{d x} [\ln (x) + 3] + \frac{d}{d x} [2]$

Step 3.2

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

$\frac{16 {(\ln (x) + 3)}^{3}}{x} + 3 {(\ln (x) + 3)}^{2} (\frac{d}{d x} [\ln (x)] + \frac{d}{d x} [3]) + \frac{d}{d x} [2]$

Step 3.3

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

$\frac{16 {(\ln (x) + 3)}^{3}}{x} + 3 {(\ln (x) + 3)}^{2} (\frac{1}{x} + \frac{d}{d x} [3]) + \frac{d}{d x} [2]$

Step 3.4

Since $3$ is constant with respect to $x$ , the derivative of $3$ with respect to $x$ is $0$ .

$\frac{16 {(\ln (x) + 3)}^{3}}{x} + 3 {(\ln (x) + 3)}^{2} (\frac{1}{x} + 0) + \frac{d}{d x} [2]$

Step 3.5

Add $\frac{1}{x}$ and $0$ .

$\frac{16 {(\ln (x) + 3)}^{3}}{x} + 3 {(\ln (x) + 3)}^{2} \frac{1}{x} + \frac{d}{d x} [2]$

Step 3.6

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

$\frac{16 {(\ln (x) + 3)}^{3}}{x} + \frac{3}{x} {(\ln (x) + 3)}^{2} + \frac{d}{d x} [2]$

Step 3.7

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

$\frac{16 {(\ln (x) + 3)}^{3}}{x} + \frac{3 {(\ln (x) + 3)}^{2}}{x} + \frac{d}{d x} [2]$

Step 4

Since $2$ is constant with respect to $x$ , the derivative of $2$ with respect to $x$ is $0$ .

$\frac{16 {(\ln (x) + 3)}^{3}}{x} + \frac{3 {(\ln (x) + 3)}^{2}}{x} + 0$

Step 5

Simplify.

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Step 5.1

Combine terms.

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Step 5.1.1

Combine the numerators over the common denominator.

$\frac{16 {(\ln (x) + 3)}^{3} + 3 {(\ln (x) + 3)}^{2}}{x} + 0$

Step 5.1.2

Add $\frac{16 {(\ln (x) + 3)}^{3} + 3 {(\ln (x) + 3)}^{2}}{x}$ and $0$ .

$\frac{16 {(\ln (x) + 3)}^{3} + 3 {(\ln (x) + 3)}^{2}}{x}$

Step 5.2

Simplify the numerator.

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Step 5.2.1

Factor ${(\ln (x) + 3)}^{2}$ out of $16 {(\ln (x) + 3)}^{3} + 3 {(\ln (x) + 3)}^{2}$ .

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Step 5.2.1.1

Factor ${(\ln (x) + 3)}^{2}$ out of $16 {(\ln (x) + 3)}^{3}$ .

$\frac{{(\ln (x) + 3)}^{2} (16 (\ln (x) + 3)) + 3 {(\ln (x) + 3)}^{2}}{x}$

Step 5.2.1.2

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

$\frac{{(\ln (x) + 3)}^{2} (16 (\ln (x) + 3)) + {(\ln (x) + 3)}^{2} \cdot 3}{x}$

Step 5.2.1.3

Factor ${(\ln (x) + 3)}^{2}$ out of ${(\ln (x) + 3)}^{2} (16 (\ln (x) + 3)) + {(\ln (x) + 3)}^{2} \cdot 3$ .

$\frac{{(\ln (x) + 3)}^{2} (16 (\ln (x) + 3) + 3)}{x}$

Step 5.2.2

Apply the distributive property.

$\frac{{(\ln (x) + 3)}^{2} (16 \ln (x) + 16 \cdot 3 + 3)}{x}$

Step 5.2.3

Multiply $16$ by $3$ .

$\frac{{(\ln (x) + 3)}^{2} (16 \ln (x) + 48 + 3)}{x}$

Step 5.2.4

Add $48$ and $3$ .

$\frac{{(\ln (x) + 3)}^{2} (16 \ln (x) + 51)}{x}$