Calculus Examples

$g (x) = 18 {(\sqrt{x + 5})}^{2} - 14$

Step 1

By the Sum Rule, the derivative of $18 {(\sqrt{x + 5})}^{2} - 14$ with respect to $x$ is $\frac{d}{d x} [18 {(\sqrt{x + 5})}^{2}] + \frac{d}{d x} [- 14]$ .

$\frac{d}{d x} [18 {(\sqrt{x + 5})}^{2}] + \frac{d}{d x} [- 14]$

Step 2

Evaluate $\frac{d}{d x} [18 {(\sqrt{x + 5})}^{2}]$ .

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Step 2.1

Rewrite ${(\sqrt{x + 5})}^{2}$ as $x + 5$ .

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Step 2.1.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{x + 5}$ as ${(x + 5)}^{\frac{1}{2}}$ .

$\frac{d}{d x} [18 {({(x + 5)}^{\frac{1}{2}})}^{2}] + \frac{d}{d x} [- 14]$

Step 2.1.2

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$\frac{d}{d x} [18 {(x + 5)}^{\frac{1}{2} \cdot 2}] + \frac{d}{d x} [- 14]$

Step 2.1.3

Combine $\frac{1}{2}$ and $2$ .

$\frac{d}{d x} [18 {(x + 5)}^{\frac{2}{2}}] + \frac{d}{d x} [- 14]$

Step 2.1.4

Cancel the common factor of $2$ .

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Step 2.1.4.1

Cancel the common factor.

$\frac{d}{d x} [18 {(x + 5)}^{\frac{2}{2}}] + \frac{d}{d x} [- 14]$

Step 2.1.4.2

Rewrite the expression.

$\frac{d}{d x} [18 {(x + 5)}^{1}] + \frac{d}{d x} [- 14]$

Step 2.1.5

Simplify.

$\frac{d}{d x} [18 (x + 5)] + \frac{d}{d x} [- 14]$

Step 2.2

Since $18$ is constant with respect to $x$ , the derivative of $18 (x + 5)$ with respect to $x$ is $18 \frac{d}{d x} [x + 5]$ .

$18 \frac{d}{d x} [x + 5] + \frac{d}{d x} [- 14]$

Step 2.3

By the Sum Rule, the derivative of $x + 5$ with respect to $x$ is $\frac{d}{d x} [x] + \frac{d}{d x} [5]$ .

$18 (\frac{d}{d x} [x] + \frac{d}{d x} [5]) + \frac{d}{d x} [- 14]$

Step 2.4

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

$18 (1 + \frac{d}{d x} [5]) + \frac{d}{d x} [- 14]$

Step 2.5

Since $5$ is constant with respect to $x$ , the derivative of $5$ with respect to $x$ is $0$ .

$18 (1 + 0) + \frac{d}{d x} [- 14]$

Step 2.6

Add $1$ and $0$ .

$18 \cdot 1 + \frac{d}{d x} [- 14]$

Step 2.7

Multiply $18$ by $1$ .

$18 + \frac{d}{d x} [- 14]$

Step 3

Differentiate using the Constant Rule.

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Step 3.1

Since $- 14$ is constant with respect to $x$ , the derivative of $- 14$ with respect to $x$ is $0$ .

$18 + 0$

Step 3.2

Add $18$ and $0$ .

$18$