Calculus Examples

$\sqrt{3 x} + 2 \cos (x)$

Step 1

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

$\frac{d}{d x} [\sqrt{3 x}] + \frac{d}{d x} [2 \cos (x)]$

Step 2

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

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Step 2.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{3 x}$ as ${(3 x)}^{\frac{1}{2}}$ .

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

Step 2.2

Factor $3$ out of $3 x$ .

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

Step 2.3

Apply the product rule to $3 (x)$ .

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

Step 2.4

Since $3^{\frac{1}{2}}$ is constant with respect to $x$ , the derivative of $3^{\frac{1}{2}} x^{\frac{1}{2}}$ with respect to $x$ is $3^{\frac{1}{2}} \frac{d}{d x} [x^{\frac{1}{2}}]$ .

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

Step 2.5

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

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

Step 2.6

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

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

Step 2.7

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

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

Step 2.8

Combine the numerators over the common denominator.

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

Step 2.9

Simplify the numerator.

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Step 2.9.1

Multiply $- 1$ by $2$ .

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

Step 2.9.2

Subtract $2$ from $1$ .

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

Step 2.10

Move the negative in front of the fraction.

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

Step 2.11

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

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

Step 2.12

Combine $3^{\frac{1}{2}}$ and $\frac{x^{- \frac{1}{2}}}{2}$ .

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

Step 2.13

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

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

Step 3

Evaluate $\frac{d}{d x} [2 \cos (x)]$ .

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Step 3.1

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

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

Step 3.2

The derivative of $\cos (x)$ with respect to $x$ is $- \sin (x)$ .

$\frac{3^{\frac{1}{2}}}{2 x^{\frac{1}{2}}} + 2 (- \sin (x))$

Step 3.3

Multiply $- 1$ by $2$ .

$\frac{3^{\frac{1}{2}}}{2 x^{\frac{1}{2}}} - 2 \sin (x)$