Calculus Examples

$y = \ln ({(x + 2)}^{5})$

Step 1

Differentiate using the chain rule, which states that $\frac{d}{d x} [f (g (x))]$ is $f' (g (x)) g' (x)$ where $f (x) = \ln (x)$ and $g (x) = {(x + 2)}^{5}$ .

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Step 1.1

To apply the Chain Rule, set $u_{1}$ as ${(x + 2)}^{5}$ .

$\frac{d}{d u_{1}} [\ln (u_{1})] \frac{d}{d x} [{(x + 2)}^{5}]$

Step 1.2

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

$\frac{1}{u_{1}} \frac{d}{d x} [{(x + 2)}^{5}]$

Step 1.3

Replace all occurrences of $u_{1}$ with ${(x + 2)}^{5}$ .

$\frac{1}{{(x + 2)}^{5}} \frac{d}{d x} [{(x + 2)}^{5}]$

Step 2

Differentiate using the chain rule, which states that $\frac{d}{d x} [f (g (x))]$ is $f' (g (x)) g' (x)$ where $f (x) = x^{5}$ and $g (x) = x + 2$ .

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Step 2.1

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

$\frac{1}{{(x + 2)}^{5}} (\frac{d}{d u_{2}} [{u_{2}}^{5}] \frac{d}{d x} [x + 2])$

Step 2.2

Differentiate using the Power Rule which states that $\frac{d}{d u_{2}} [{u_{2}}^{n}]$ is $n {u_{2}}^{n - 1}$ where $n = 5$ .

$\frac{1}{{(x + 2)}^{5}} (5 {u_{2}}^{4} \frac{d}{d x} [x + 2])$

Step 2.3

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

$\frac{1}{{(x + 2)}^{5}} (5 {(x + 2)}^{4} \frac{d}{d x} [x + 2])$

Step 3

Differentiate.

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Step 3.1

Combine $5$ and $\frac{1}{{(x + 2)}^{5}}$ .

$\frac{5}{{(x + 2)}^{5}} ({(x + 2)}^{4} \frac{d}{d x} [x + 2])$

Step 3.2

Simplify terms.

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Step 3.2.1

Combine ${(x + 2)}^{4}$ and $\frac{5}{{(x + 2)}^{5}}$ .

$\frac{{(x + 2)}^{4} \cdot 5}{{(x + 2)}^{5}} \frac{d}{d x} [x + 2]$

Step 3.2.2

Move $5$ to the left of ${(x + 2)}^{4}$ .

$\frac{5 \cdot {(x + 2)}^{4}}{{(x + 2)}^{5}} \frac{d}{d x} [x + 2]$

Step 3.2.3

Cancel the common factor of ${(x + 2)}^{4}$ and ${(x + 2)}^{5}$ .

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Step 3.2.3.1

Factor ${(x + 2)}^{4}$ out of $5 {(x + 2)}^{4}$ .

$\frac{{(x + 2)}^{4} \cdot 5}{{(x + 2)}^{5}} \frac{d}{d x} [x + 2]$

Step 3.2.3.2

Cancel the common factors.

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Step 3.2.3.2.1

Factor ${(x + 2)}^{4}$ out of ${(x + 2)}^{5}$ .

$\frac{{(x + 2)}^{4} \cdot 5}{{(x + 2)}^{4} (x + 2)} \frac{d}{d x} [x + 2]$

Step 3.2.3.2.2

Cancel the common factor.

$\frac{{(x + 2)}^{4} \cdot 5}{{(x + 2)}^{4} (x + 2)} \frac{d}{d x} [x + 2]$

Step 3.2.3.2.3

Rewrite the expression.

$\frac{5}{x + 2} \frac{d}{d x} [x + 2]$

Step 3.3

By the Sum Rule, the derivative of $x + 2$ with respect to $x$ is $\frac{d}{d x} [x] + \frac{d}{d x} [2]$ .

$\frac{5}{x + 2} (\frac{d}{d x} [x] + \frac{d}{d x} [2])$

Step 3.4

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

$\frac{5}{x + 2} (1 + \frac{d}{d x} [2])$

Step 3.5

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

$\frac{5}{x + 2} (1 + 0)$

Step 3.6

Simplify the expression.

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Step 3.6.1

Add $1$ and $0$ .

$\frac{5}{x + 2} \cdot 1$

Step 3.6.2

Multiply $\frac{5}{x + 2}$ by $1$ .

$\frac{5}{x + 2}$