Calculus Examples

$f (x) = e^{\sqrt{x} + 21} + e^{p}$

Step 1

By the Sum Rule, the derivative of $e^{\sqrt{x} + 21} + e^{p}$ with respect to $x$ is $\frac{d}{d x} [e^{\sqrt{x} + 21}] + \frac{d}{d x} [e^{p}]$ .

$\frac{d}{d x} [e^{\sqrt{x} + 21}] + \frac{d}{d x} [e^{p}]$

Step 2

Evaluate $\frac{d}{d x} [e^{\sqrt{x} + 21}]$ .

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Step 2.1

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

$\frac{d}{d x} [e^{x^{\frac{1}{2}} + 21}] + \frac{d}{d x} [e^{p}]$

Step 2.2

Differentiate using the chain rule, which states that $\frac{d}{d x} [f (g (x))]$ is $f' (g (x)) g' (x)$ where $f (x) = e^{x}$ and $g (x) = x^{\frac{1}{2}} + 21$ .

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Step 2.2.1

To apply the Chain Rule, set $u$ as $x^{\frac{1}{2}} + 21$ .

$\frac{d}{d u} [e^{u}] \frac{d}{d x} [x^{\frac{1}{2}} + 21] + \frac{d}{d x} [e^{p}]$

Step 2.2.2

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

$e^{u} \frac{d}{d x} [x^{\frac{1}{2}} + 21] + \frac{d}{d x} [e^{p}]$

Step 2.2.3

Replace all occurrences of $u$ with $x^{\frac{1}{2}} + 21$ .

$e^{x^{\frac{1}{2}} + 21} \frac{d}{d x} [x^{\frac{1}{2}} + 21] + \frac{d}{d x} [e^{p}]$

Step 2.3

By the Sum Rule, the derivative of $x^{\frac{1}{2}} + 21$ with respect to $x$ is $\frac{d}{d x} [x^{\frac{1}{2}}] + \frac{d}{d x} [21]$ .

$e^{x^{\frac{1}{2}} + 21} (\frac{d}{d x} [x^{\frac{1}{2}}] + \frac{d}{d x} [21]) + \frac{d}{d x} [e^{p}]$

Step 2.4

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

$e^{x^{\frac{1}{2}} + 21} (\frac{1}{2} x^{\frac{1}{2} - 1} + \frac{d}{d x} [21]) + \frac{d}{d x} [e^{p}]$

Step 2.5

Since $21$ is constant with respect to $x$ , the derivative of $21$ with respect to $x$ is $0$ .

$e^{x^{\frac{1}{2}} + 21} (\frac{1}{2} x^{\frac{1}{2} - 1} + 0) + \frac{d}{d x} [e^{p}]$

Step 2.6

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

$e^{x^{\frac{1}{2}} + 21} (\frac{1}{2} x^{\frac{1}{2} - 1 \cdot \frac{2}{2}} + 0) + \frac{d}{d x} [e^{p}]$

Step 2.7

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

$e^{x^{\frac{1}{2}} + 21} (\frac{1}{2} x^{\frac{1}{2} + \frac{- 1 \cdot 2}{2}} + 0) + \frac{d}{d x} [e^{p}]$

Step 2.8

Combine the numerators over the common denominator.

$e^{x^{\frac{1}{2}} + 21} (\frac{1}{2} x^{\frac{1 - 1 \cdot 2}{2}} + 0) + \frac{d}{d x} [e^{p}]$

Step 2.9

Simplify the numerator.

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Step 2.9.1

Multiply $- 1$ by $2$ .

$e^{x^{\frac{1}{2}} + 21} (\frac{1}{2} x^{\frac{1 - 2}{2}} + 0) + \frac{d}{d x} [e^{p}]$

Step 2.9.2

Subtract $2$ from $1$ .

$e^{x^{\frac{1}{2}} + 21} (\frac{1}{2} x^{\frac{- 1}{2}} + 0) + \frac{d}{d x} [e^{p}]$

Step 2.10

Move the negative in front of the fraction.

$e^{x^{\frac{1}{2}} + 21} (\frac{1}{2} x^{- \frac{1}{2}} + 0) + \frac{d}{d x} [e^{p}]$

Step 2.11

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

$e^{x^{\frac{1}{2}} + 21} (\frac{1}{2} x^{- \frac{1}{2}}) + \frac{d}{d x} [e^{p}]$

Step 2.12

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

$e^{x^{\frac{1}{2}} + 21} \frac{x^{- \frac{1}{2}}}{2} + \frac{d}{d x} [e^{p}]$

Step 2.13

Combine $e^{x^{\frac{1}{2}} + 21}$ and $\frac{x^{- \frac{1}{2}}}{2}$ .

$\frac{e^{x^{\frac{1}{2}} + 21} x^{- \frac{1}{2}}}{2} + \frac{d}{d x} [e^{p}]$

Step 2.14

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

$\frac{e^{x^{\frac{1}{2}} + 21}}{2 x^{\frac{1}{2}}} + \frac{d}{d x} [e^{p}]$

Step 3

Differentiate using the Constant Rule.

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Step 3.1

Since $e^{p}$ is constant with respect to $x$ , the derivative of $e^{p}$ with respect to $x$ is $0$ .

$\frac{e^{x^{\frac{1}{2}} + 21}}{2 x^{\frac{1}{2}}} + 0$

Step 3.2

Add $\frac{e^{x^{\frac{1}{2}} + 21}}{2 x^{\frac{1}{2}}}$ and $0$ .

$\frac{e^{x^{\frac{1}{2}} + 21}}{2 x^{\frac{1}{2}}}$