Calculus Examples

$f (x) = 8 x + 7 \sqrt{x}$

Step 1

By the Sum Rule, the derivative of $8 x + 7 \sqrt{x}$ with respect to $x$ is $\frac{d}{d x} [8 x] + \frac{d}{d x} [7 \sqrt{x}]$ .

$\frac{d}{d x} [8 x] + \frac{d}{d x} [7 \sqrt{x}]$

Step 2

Evaluate $\frac{d}{d x} [8 x]$ .

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Step 2.1

Since $8$ is constant with respect to $x$ , the derivative of $8 x$ with respect to $x$ is $8 \frac{d}{d x} [x]$ .

$8 \frac{d}{d x} [x] + \frac{d}{d x} [7 \sqrt{x}]$

Step 2.2

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

$8 \cdot 1 + \frac{d}{d x} [7 \sqrt{x}]$

Step 2.3

Multiply $8$ by $1$ .

$8 + \frac{d}{d x} [7 \sqrt{x}]$

Step 3

Evaluate $\frac{d}{d x} [7 \sqrt{x}]$ .

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Step 3.1

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

$8 + \frac{d}{d x} [7 x^{\frac{1}{2}}]$

Step 3.2

Since $7$ is constant with respect to $x$ , the derivative of $7 x^{\frac{1}{2}}$ with respect to $x$ is $7 \frac{d}{d x} [x^{\frac{1}{2}}]$ .

$8 + 7 \frac{d}{d x} [x^{\frac{1}{2}}]$

Step 3.3

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

$8 + 7 (\frac{1}{2} x^{\frac{1}{2} - 1})$

Step 3.4

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

$8 + 7 (\frac{1}{2} x^{\frac{1}{2} - 1 \cdot \frac{2}{2}})$

Step 3.5

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

$8 + 7 (\frac{1}{2} x^{\frac{1}{2} + \frac{- 1 \cdot 2}{2}})$

Step 3.6

Combine the numerators over the common denominator.

$8 + 7 (\frac{1}{2} x^{\frac{1 - 1 \cdot 2}{2}})$

Step 3.7

Simplify the numerator.

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Step 3.7.1

Multiply $- 1$ by $2$ .

$8 + 7 (\frac{1}{2} x^{\frac{1 - 2}{2}})$

Step 3.7.2

Subtract $2$ from $1$ .

$8 + 7 (\frac{1}{2} x^{\frac{- 1}{2}})$

Step 3.8

Move the negative in front of the fraction.

$8 + 7 (\frac{1}{2} x^{- \frac{1}{2}})$

Step 3.9

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

$8 + 7 \frac{x^{- \frac{1}{2}}}{2}$

Step 3.10

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

$8 + \frac{7 x^{- \frac{1}{2}}}{2}$

Step 3.11

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

$8 + \frac{7}{2 x^{\frac{1}{2}}}$