Calculus Examples

$7^{\sqrt{x} + 2}$

Step 1

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

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

Step 2

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

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Step 2.1

To apply the Chain Rule, set $u$ as $x^{\frac{1}{2}} + 2$ .

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

Step 2.2

Differentiate using the Exponential Rule which states that $\frac{d}{d u} [a^{u}]$ is $a^{u} \ln (a)$ where $a$ = $7$ .

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

Step 2.3

Replace all occurrences of $u$ with $x^{\frac{1}{2}} + 2$ .

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

Step 3

Differentiate.

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Step 3.1

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

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

Step 3.2

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

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

Step 4

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

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

Step 5

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

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

Step 6

Combine the numerators over the common denominator.

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

Step 7

Simplify the numerator.

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Step 7.1

Multiply $- 1$ by $2$ .

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

Step 7.2

Subtract $2$ from $1$ .

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

Step 8

Combine fractions.

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Step 8.1

Move the negative in front of the fraction.

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

Step 8.2

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

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

Step 8.3

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

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

Step 9

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

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

Step 10

Combine fractions.

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Step 10.1

Add $\frac{1}{2 x^{\frac{1}{2}}}$ and $0$ .

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

Step 10.2

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

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

Step 10.3

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

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