Calculus Examples

$2 e^{\sqrt{x}}$

Step 1

Differentiate using the Constant Multiple Rule.

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Step 1.1

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

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

Step 1.2

Since $2$ is constant with respect to $x$ , the derivative of $2 e^{x^{\frac{1}{2}}}$ with respect to $x$ is $2 \frac{d}{d x} [e^{x^{\frac{1}{2}}}]$ .

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

Step 2

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

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Step 2.1

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

$2 (\frac{d}{d u} [e^{u}] \frac{d}{d x} [x^{\frac{1}{2}}])$

Step 2.2

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

$2 (e^{u} \frac{d}{d x} [x^{\frac{1}{2}}])$

Step 2.3

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

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

Step 3

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

$2 e^{x^{\frac{1}{2}}} (\frac{1}{2} x^{\frac{1}{2} - 1})$

Step 4

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

$2 e^{x^{\frac{1}{2}}} (\frac{1}{2} x^{\frac{1}{2} - 1 \cdot \frac{2}{2}})$

Step 5

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

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

Step 6

Combine the numerators over the common denominator.

$2 e^{x^{\frac{1}{2}}} (\frac{1}{2} x^{\frac{1 - 1 \cdot 2}{2}})$

Step 7

Simplify the numerator.

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Step 7.1

Multiply $- 1$ by $2$ .

$2 e^{x^{\frac{1}{2}}} (\frac{1}{2} x^{\frac{1 - 2}{2}})$

Step 7.2

Subtract $2$ from $1$ .

$2 e^{x^{\frac{1}{2}}} (\frac{1}{2} x^{\frac{- 1}{2}})$

Step 8

Move the negative in front of the fraction.

$2 e^{x^{\frac{1}{2}}} (\frac{1}{2} x^{- \frac{1}{2}})$

Step 9

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

$2 e^{x^{\frac{1}{2}}} \frac{x^{- \frac{1}{2}}}{2}$

Step 10

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

$\frac{x^{- \frac{1}{2}} \cdot 2}{2} e^{x^{\frac{1}{2}}}$

Step 11

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

$\frac{x^{- \frac{1}{2}} \cdot 2 e^{x^{\frac{1}{2}}}}{2}$

Step 12

Simplify the expression.

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Step 12.1

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

$\frac{2 \cdot x^{- \frac{1}{2}} e^{x^{\frac{1}{2}}}}{2}$

Step 12.2

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

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

Step 13

Cancel the common factor.

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

Step 14

Rewrite the expression.

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