Calculus Examples

$C (20) = 9 + \sqrt{2 x + 24}$

Step 1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{2 x + 24}$ as ${(2 x + 24)}^{\frac{1}{2}}$ .

$C (20) = 9 + {(2 x + 24)}^{\frac{1}{2}}$

Step 2

Differentiate both sides of the equation.

$\frac{d}{d x} (C (20)) = \frac{d}{d x} (9 + {(2 x + 24)}^{\frac{1}{2}})$

Step 3

Differentiate the left side of the equation.

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Step 3.1

Move $20$ to the left of $C$ .

$\frac{d}{d x} [20 C]$

Step 3.2

Since $20$ is constant with respect to $x$ , the derivative of $20 C$ with respect to $x$ is $20 \frac{d}{d x} [C]$ .

$20 \frac{d}{d x} [C]$

Step 3.3

Rewrite $\frac{d}{d x} [C]$ as $C'$ .

$20 C'$

Step 4

Differentiate the right side of the equation.

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Step 4.1

Differentiate.

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Step 4.1.1

By the Sum Rule, the derivative of $9 + {(2 x + 24)}^{\frac{1}{2}}$ with respect to $x$ is $\frac{d}{d x} [9] + \frac{d}{d x} [{(2 x + 24)}^{\frac{1}{2}}]$ .

$\frac{d}{d x} [9] + \frac{d}{d x} [{(2 x + 24)}^{\frac{1}{2}}]$

Step 4.1.2

Since $9$ is constant with respect to $x$ , the derivative of $9$ with respect to $x$ is $0$ .

$0 + \frac{d}{d x} [{(2 x + 24)}^{\frac{1}{2}}]$

Step 4.2

Evaluate $\frac{d}{d x} [{(2 x + 24)}^{\frac{1}{2}}]$ .

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Step 4.2.1

Differentiate using the chain rule, which states that $\frac{d}{d x} [f (g (x))]$ is $f' (g (x)) g' (x)$ where $f (x) = x^{\frac{1}{2}}$ and $g (x) = 2 x + 24$ .

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Step 4.2.1.1

To apply the Chain Rule, set $u$ as $2 x + 24$ .

$0 + \frac{d}{d u} [u^{\frac{1}{2}}] \frac{d}{d x} [2 x + 24]$

Step 4.2.1.2

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

$0 + \frac{1}{2} u^{\frac{1}{2} - 1} \frac{d}{d x} [2 x + 24]$

Step 4.2.1.3

Replace all occurrences of $u$ with $2 x + 24$ .

$0 + \frac{1}{2} {(2 x + 24)}^{\frac{1}{2} - 1} \frac{d}{d x} [2 x + 24]$

Step 4.2.2

By the Sum Rule, the derivative of $2 x + 24$ with respect to $x$ is $\frac{d}{d x} [2 x] + \frac{d}{d x} [24]$ .

$0 + \frac{1}{2} {(2 x + 24)}^{\frac{1}{2} - 1} (\frac{d}{d x} [2 x] + \frac{d}{d x} [24])$

Step 4.2.3

Since $2$ is constant with respect to $x$ , the derivative of $2 x$ with respect to $x$ is $2 \frac{d}{d x} [x]$ .

$0 + \frac{1}{2} {(2 x + 24)}^{\frac{1}{2} - 1} (2 \frac{d}{d x} [x] + \frac{d}{d x} [24])$

Step 4.2.4

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

$0 + \frac{1}{2} {(2 x + 24)}^{\frac{1}{2} - 1} (2 \cdot 1 + \frac{d}{d x} [24])$

Step 4.2.5

Since $24$ is constant with respect to $x$ , the derivative of $24$ with respect to $x$ is $0$ .

$0 + \frac{1}{2} {(2 x + 24)}^{\frac{1}{2} - 1} (2 \cdot 1 + 0)$

Step 4.2.6

To write $- 1$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$0 + \frac{1}{2} {(2 x + 24)}^{\frac{1}{2} - 1 \cdot \frac{2}{2}} (2 \cdot 1 + 0)$

Step 4.2.7

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

$0 + \frac{1}{2} {(2 x + 24)}^{\frac{1}{2} + \frac{- 1 \cdot 2}{2}} (2 \cdot 1 + 0)$

Step 4.2.8

Combine the numerators over the common denominator.

$0 + \frac{1}{2} {(2 x + 24)}^{\frac{1 - 1 \cdot 2}{2}} (2 \cdot 1 + 0)$

Step 4.2.9

Simplify the numerator.

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Step 4.2.9.1

Multiply $- 1$ by $2$ .

$0 + \frac{1}{2} {(2 x + 24)}^{\frac{1 - 2}{2}} (2 \cdot 1 + 0)$

Step 4.2.9.2

Subtract $2$ from $1$ .

$0 + \frac{1}{2} {(2 x + 24)}^{\frac{- 1}{2}} (2 \cdot 1 + 0)$

Step 4.2.10

Move the negative in front of the fraction.

$0 + \frac{1}{2} {(2 x + 24)}^{- \frac{1}{2}} (2 \cdot 1 + 0)$

Step 4.2.11

Multiply $2$ by $1$ .

$0 + \frac{1}{2} {(2 x + 24)}^{- \frac{1}{2}} (2 + 0)$

Step 4.2.12

Add $2$ and $0$ .

$0 + \frac{1}{2} {(2 x + 24)}^{- \frac{1}{2}} \cdot 2$

Step 4.2.13

Combine $\frac{1}{2}$ and ${(2 x + 24)}^{- \frac{1}{2}}$ .

$0 + \frac{{(2 x + 24)}^{- \frac{1}{2}}}{2} \cdot 2$

Step 4.2.14

Combine $\frac{{(2 x + 24)}^{- \frac{1}{2}}}{2}$ and $2$ .

$0 + \frac{{(2 x + 24)}^{- \frac{1}{2}} \cdot 2}{2}$

Step 4.2.15

Move $2$ to the left of ${(2 x + 24)}^{- \frac{1}{2}}$ .

$0 + \frac{2 \cdot {(2 x + 24)}^{- \frac{1}{2}}}{2}$

Step 4.2.16

Move ${(2 x + 24)}^{- \frac{1}{2}}$ to the denominator using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$0 + \frac{2}{2 {(2 x + 24)}^{\frac{1}{2}}}$

Step 4.2.17

Cancel the common factor.

$0 + \frac{2}{2 {(2 x + 24)}^{\frac{1}{2}}}$

Step 4.2.18

Rewrite the expression.

$0 + \frac{1}{{(2 x + 24)}^{\frac{1}{2}}}$

Step 4.3

Add $0$ and $\frac{1}{{(2 x + 24)}^{\frac{1}{2}}}$ .

$\frac{1}{{(2 x + 24)}^{\frac{1}{2}}}$

Step 5

Reform the equation by setting the left side equal to the right side.

$20 C' = \frac{1}{{(2 x + 24)}^{\frac{1}{2}}}$

Step 6

Divide each term in $20 C' = \frac{1}{{(2 x + 24)}^{\frac{1}{2}}}$ by $20$ and simplify.

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Step 6.1

Divide each term in $20 C' = \frac{1}{{(2 x + 24)}^{\frac{1}{2}}}$ by $20$ .

$\frac{20 C'}{20} = \frac{\frac{1}{{(2 x + 24)}^{\frac{1}{2}}}}{20}$

Step 6.2

Simplify the left side.

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Step 6.2.1

Cancel the common factor of $20$ .

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Step 6.2.1.1

Cancel the common factor.

$\frac{20 C'}{20} = \frac{\frac{1}{{(2 x + 24)}^{\frac{1}{2}}}}{20}$

Step 6.2.1.2

Divide $C'$ by $1$ .

$C' = \frac{\frac{1}{{(2 x + 24)}^{\frac{1}{2}}}}{20}$

Step 6.3

Simplify the right side.

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Step 6.3.1

Multiply the numerator by the reciprocal of the denominator.

$C' = \frac{1}{{(2 x + 24)}^{\frac{1}{2}}} \cdot \frac{1}{20}$

Step 6.3.2

Combine.

$C' = \frac{1 \cdot 1}{{(2 x + 24)}^{\frac{1}{2}} \cdot 20}$

Step 6.3.3

Simplify the expression.

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Step 6.3.3.1

Multiply $1$ by $1$ .

$C' = \frac{1}{{(2 x + 24)}^{\frac{1}{2}} \cdot 20}$

Step 6.3.3.2

Move $20$ to the left of ${(2 x + 24)}^{\frac{1}{2}}$ .

$C' = \frac{1}{20 {(2 x + 24)}^{\frac{1}{2}}}$

Step 7

Replace $C'$ with $\frac{d C}{d x}$ .

$\frac{d C}{d x} = \frac{1}{20 {(2 x + 24)}^{\frac{1}{2}}}$