Calculus Examples

$y = \ln (\sqrt{x + 7})$

Step 1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{x + 7}$ as ${(x + 7)}^{\frac{1}{2}}$ .

$y = \ln ({(x + 7)}^{\frac{1}{2}})$

Step 2

Differentiate both sides of the equation.

$\frac{d}{d x} (y) = \frac{d}{d x} (\ln ({(x + 7)}^{\frac{1}{2}}))$

Step 3

The derivative of $y$ with respect to $x$ is $y'$ .

$y'$

Step 4

Differentiate the right side of the equation.

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Step 4.1

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 $g (x) = {(x + 7)}^{\frac{1}{2}}$ .

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Step 4.1.1

To apply the Chain Rule, set $u_{1}$ as ${(x + 7)}^{\frac{1}{2}}$ .

$\frac{d}{d u_{1}} [\ln (u_{1})] \frac{d}{d x} [{(x + 7)}^{\frac{1}{2}}]$

Step 4.1.2

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

$\frac{1}{u_{1}} \frac{d}{d x} [{(x + 7)}^{\frac{1}{2}}]$

Step 4.1.3

Replace all occurrences of $u_{1}$ with ${(x + 7)}^{\frac{1}{2}}$ .

$\frac{1}{{(x + 7)}^{\frac{1}{2}}} \frac{d}{d x} [{(x + 7)}^{\frac{1}{2}}]$

Step 4.2

Differentiate using the chain rule, which states that $\frac{d}{d x} [f (g (x))]$ is $f' (g (x)) g' (x)$ where $f (x) = x^{\frac{1}{2}}$ and $g (x) = x + 7$ .

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Step 4.2.1

To apply the Chain Rule, set $u_{2}$ as $x + 7$ .

$\frac{1}{{(x + 7)}^{\frac{1}{2}}} (\frac{d}{d u_{2}} [{u_{2}}^{\frac{1}{2}}] \frac{d}{d x} [x + 7])$

Step 4.2.2

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

$\frac{1}{{(x + 7)}^{\frac{1}{2}}} (\frac{1}{2} {u_{2}}^{\frac{1}{2} - 1} \frac{d}{d x} [x + 7])$

Step 4.2.3

Replace all occurrences of $u_{2}$ with $x + 7$ .

$\frac{1}{{(x + 7)}^{\frac{1}{2}}} (\frac{1}{2} {(x + 7)}^{\frac{1}{2} - 1} \frac{d}{d x} [x + 7])$

Step 4.3

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

$\frac{1}{{(x + 7)}^{\frac{1}{2}}} (\frac{1}{2} {(x + 7)}^{\frac{1}{2} - 1 \cdot \frac{2}{2}} \frac{d}{d x} [x + 7])$

Step 4.4

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

$\frac{1}{{(x + 7)}^{\frac{1}{2}}} (\frac{1}{2} {(x + 7)}^{\frac{1}{2} + \frac{- 1 \cdot 2}{2}} \frac{d}{d x} [x + 7])$

Step 4.5

Combine the numerators over the common denominator.

$\frac{1}{{(x + 7)}^{\frac{1}{2}}} (\frac{1}{2} {(x + 7)}^{\frac{1 - 1 \cdot 2}{2}} \frac{d}{d x} [x + 7])$

Step 4.6

Simplify the numerator.

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Step 4.6.1

Multiply $- 1$ by $2$ .

$\frac{1}{{(x + 7)}^{\frac{1}{2}}} (\frac{1}{2} {(x + 7)}^{\frac{1 - 2}{2}} \frac{d}{d x} [x + 7])$

Step 4.6.2

Subtract $2$ from $1$ .

$\frac{1}{{(x + 7)}^{\frac{1}{2}}} (\frac{1}{2} {(x + 7)}^{\frac{- 1}{2}} \frac{d}{d x} [x + 7])$

Step 4.7

Move the negative in front of the fraction.

$\frac{1}{{(x + 7)}^{\frac{1}{2}}} (\frac{1}{2} {(x + 7)}^{- \frac{1}{2}} \frac{d}{d x} [x + 7])$

Step 4.8

Combine $\frac{1}{2}$ and ${(x + 7)}^{- \frac{1}{2}}$ .

$\frac{1}{{(x + 7)}^{\frac{1}{2}}} (\frac{{(x + 7)}^{- \frac{1}{2}}}{2} \frac{d}{d x} [x + 7])$

Step 4.9

Move ${(x + 7)}^{- \frac{1}{2}}$ to the denominator using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$\frac{1}{{(x + 7)}^{\frac{1}{2}}} (\frac{1}{2 {(x + 7)}^{\frac{1}{2}}} \frac{d}{d x} [x + 7])$

Step 4.10

Multiply $\frac{1}{2 {(x + 7)}^{\frac{1}{2}}}$ by $\frac{1}{{(x + 7)}^{\frac{1}{2}}}$ .

$\frac{1}{2 {(x + 7)}^{\frac{1}{2}} {(x + 7)}^{\frac{1}{2}}} \frac{d}{d x} [x + 7]$

Step 4.11

Multiply ${(x + 7)}^{\frac{1}{2}}$ by ${(x + 7)}^{\frac{1}{2}}$ by adding the exponents.

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Step 4.11.1

Move ${(x + 7)}^{\frac{1}{2}}$ .

$\frac{1}{2 ({(x + 7)}^{\frac{1}{2}} {(x + 7)}^{\frac{1}{2}})} \frac{d}{d x} [x + 7]$

Step 4.11.2

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$\frac{1}{2 {(x + 7)}^{\frac{1}{2} + \frac{1}{2}}} \frac{d}{d x} [x + 7]$

Step 4.11.3

Combine the numerators over the common denominator.

$\frac{1}{2 {(x + 7)}^{\frac{1 + 1}{2}}} \frac{d}{d x} [x + 7]$

Step 4.11.4

Add $1$ and $1$ .

$\frac{1}{2 {(x + 7)}^{\frac{2}{2}}} \frac{d}{d x} [x + 7]$

Step 4.11.5

Divide $2$ by $2$ .

$\frac{1}{2 {(x + 7)}^{1}} \frac{d}{d x} [x + 7]$

Step 4.12

Simplify $2 {(x + 7)}^{1}$ .

$\frac{1}{2 (x + 7)} \frac{d}{d x} [x + 7]$

Step 4.13

By the Sum Rule, the derivative of $x + 7$ with respect to $x$ is $\frac{d}{d x} [x] + \frac{d}{d x} [7]$ .

$\frac{1}{2 (x + 7)} (\frac{d}{d x} [x] + \frac{d}{d x} [7])$

Step 4.14

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

$\frac{1}{2 (x + 7)} (1 + \frac{d}{d x} [7])$

Step 4.15

Since $7$ is constant with respect to $x$ , the derivative of $7$ with respect to $x$ is $0$ .

$\frac{1}{2 (x + 7)} (1 + 0)$

Step 4.16

Simplify the expression.

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Step 4.16.1

Add $1$ and $0$ .

$\frac{1}{2 (x + 7)} \cdot 1$

Step 4.16.2

Multiply $\frac{1}{2 (x + 7)}$ by $1$ .

$\frac{1}{2 (x + 7)}$

Step 5

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

$y' = \frac{1}{2 (x + 7)}$

Step 6

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

$\frac{d y}{d x} = \frac{1}{2 (x + 7)}$