Calculus Examples

$f (x) = 2 \sqrt{x} + 3 e^{x}$

Step 1

By the Sum Rule, the derivative of $2 \sqrt{x} + 3 e^{x}$ with respect to $x$ is $\frac{d}{d x} [2 \sqrt{x}] + \frac{d}{d x} [3 e^{x}]$ .

$\frac{d}{d x} [2 \sqrt{x}] + \frac{d}{d x} [3 e^{x}]$

Step 2

Evaluate $\frac{d}{d x} [2 \sqrt{x}]$ .

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Step 2.1

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

$\frac{d}{d x} [2 x^{\frac{1}{2}}] + \frac{d}{d x} [3 e^{x}]$

Step 2.2

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

$2 \frac{d}{d x} [x^{\frac{1}{2}}] + \frac{d}{d x} [3 e^{x}]$

Step 2.3

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

$2 (\frac{1}{2} x^{\frac{1}{2} - 1}) + \frac{d}{d x} [3 e^{x}]$

Step 2.4

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

$2 (\frac{1}{2} x^{\frac{1}{2} - 1 \cdot \frac{2}{2}}) + \frac{d}{d x} [3 e^{x}]$

Step 2.5

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

$2 (\frac{1}{2} x^{\frac{1}{2} + \frac{- 1 \cdot 2}{2}}) + \frac{d}{d x} [3 e^{x}]$

Step 2.6

Combine the numerators over the common denominator.

$2 (\frac{1}{2} x^{\frac{1 - 1 \cdot 2}{2}}) + \frac{d}{d x} [3 e^{x}]$

Step 2.7

Simplify the numerator.

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Step 2.7.1

Multiply $- 1$ by $2$ .

$2 (\frac{1}{2} x^{\frac{1 - 2}{2}}) + \frac{d}{d x} [3 e^{x}]$

Step 2.7.2

Subtract $2$ from $1$ .

$2 (\frac{1}{2} x^{\frac{- 1}{2}}) + \frac{d}{d x} [3 e^{x}]$

Step 2.8

Move the negative in front of the fraction.

$2 (\frac{1}{2} x^{- \frac{1}{2}}) + \frac{d}{d x} [3 e^{x}]$

Step 2.9

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

$2 \frac{x^{- \frac{1}{2}}}{2} + \frac{d}{d x} [3 e^{x}]$

Step 2.10

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

$\frac{2 x^{- \frac{1}{2}}}{2} + \frac{d}{d x} [3 e^{x}]$

Step 2.11

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

$\frac{2}{2 x^{\frac{1}{2}}} + \frac{d}{d x} [3 e^{x}]$

Step 2.12

Cancel the common factor.

$\frac{2}{2 x^{\frac{1}{2}}} + \frac{d}{d x} [3 e^{x}]$

Step 2.13

Rewrite the expression.

$\frac{1}{x^{\frac{1}{2}}} + \frac{d}{d x} [3 e^{x}]$

Step 3

Evaluate $\frac{d}{d x} [3 e^{x}]$ .

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Step 3.1

Since $3$ is constant with respect to $x$ , the derivative of $3 e^{x}$ with respect to $x$ is $3 \frac{d}{d x} [e^{x}]$ .

$\frac{1}{x^{\frac{1}{2}}} + 3 \frac{d}{d x} [e^{x}]$

Step 3.2

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

$\frac{1}{x^{\frac{1}{2}}} + 3 e^{x}$

Step 4

Reorder terms.

$3 e^{x} + \frac{1}{x^{\frac{1}{2}}}$