Calculus Examples

$100 (7 + 3 \sqrt{x})$

Step 1

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

$\frac{d}{d x} [100 (7 + 3 x^{\frac{1}{2}})]$

Step 2

Since $100$ is constant with respect to $x$ , the derivative of $100 (7 + 3 x^{\frac{1}{2}})$ with respect to $x$ is $100 \frac{d}{d x} [7 + 3 x^{\frac{1}{2}}]$ .

$100 \frac{d}{d x} [7 + 3 x^{\frac{1}{2}}]$

Step 3

By the Sum Rule, the derivative of $7 + 3 x^{\frac{1}{2}}$ with respect to $x$ is $\frac{d}{d x} [7] + \frac{d}{d x} [3 x^{\frac{1}{2}}]$ .

$100 (\frac{d}{d x} [7] + \frac{d}{d x} [3 x^{\frac{1}{2}}])$

Step 4

Since $7$ is constant with respect to $x$ , the derivative of $7$ with respect to $x$ is $0$ .

$100 (0 + \frac{d}{d x} [3 x^{\frac{1}{2}}])$

Step 5

Add $0$ and $\frac{d}{d x} [3 x^{\frac{1}{2}}]$ .

$100 \frac{d}{d x} [3 x^{\frac{1}{2}}]$

Step 6

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

$100 (3 \frac{d}{d x} [x^{\frac{1}{2}}])$

Step 7

Multiply $3$ by $100$ .

$300 \frac{d}{d x} [x^{\frac{1}{2}}]$

Step 8

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

$300 (\frac{1}{2} x^{\frac{1}{2} - 1})$

Step 9

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

$300 (\frac{1}{2} x^{\frac{1}{2} - 1 \cdot \frac{2}{2}})$

Step 10

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

$300 (\frac{1}{2} x^{\frac{1}{2} + \frac{- 1 \cdot 2}{2}})$

Step 11

Combine the numerators over the common denominator.

$300 (\frac{1}{2} x^{\frac{1 - 1 \cdot 2}{2}})$

Step 12

Simplify the numerator.

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Step 12.1

Multiply $- 1$ by $2$ .

$300 (\frac{1}{2} x^{\frac{1 - 2}{2}})$

Step 12.2

Subtract $2$ from $1$ .

$300 (\frac{1}{2} x^{\frac{- 1}{2}})$

Step 13

Move the negative in front of the fraction.

$300 (\frac{1}{2} x^{- \frac{1}{2}})$

Step 14

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

$300 \frac{x^{- \frac{1}{2}}}{2}$

Step 15

Combine $300$ and $\frac{x^{- \frac{1}{2}}}{2}$ .

$\frac{300 x^{- \frac{1}{2}}}{2}$

Step 16

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

$\frac{300}{2 x^{\frac{1}{2}}}$

Step 17

Factor $2$ out of $300$ .

$\frac{2 \cdot 150}{2 x^{\frac{1}{2}}}$

Step 18

Cancel the common factors.

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Step 18.1

Factor $2$ out of $2 x^{\frac{1}{2}}$ .

$\frac{2 \cdot 150}{2 (x^{\frac{1}{2}})}$

Step 18.2

Cancel the common factor.

$\frac{2 \cdot 150}{2 x^{\frac{1}{2}}}$

Step 18.3

Rewrite the expression.

$\frac{150}{x^{\frac{1}{2}}}$