Calculus Examples

$(\frac{50}{\sqrt{x}}) x$

Step 1

Combine $\frac{50}{\sqrt{x}}$ and $x$ .

$\frac{d}{d x} [\frac{50 x}{\sqrt{x}}]$

Step 2

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

$\frac{d}{d x} [\frac{50 x}{x^{\frac{1}{2}}}]$

Step 3

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

$\frac{d}{d x} [50 x \cdot x^{- \frac{1}{2}}]$

Step 4

Multiply $x$ by $x^{- \frac{1}{2}}$ by adding the exponents.

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Step 4.1

Move $x^{- \frac{1}{2}}$ .

$\frac{d}{d x} [50 (x^{- \frac{1}{2}} x)]$

Step 4.2

Multiply $x^{- \frac{1}{2}}$ by $x$ .

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Step 4.2.1

Raise $x$ to the power of $1$ .

$\frac{d}{d x} [50 (x^{- \frac{1}{2}} x^{1})]$

Step 4.2.2

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$\frac{d}{d x} [50 x^{- \frac{1}{2} + 1}]$

Step 4.3

Write $1$ as a fraction with a common denominator.

$\frac{d}{d x} [50 x^{- \frac{1}{2} + \frac{2}{2}}]$

Step 4.4

Combine the numerators over the common denominator.

$\frac{d}{d x} [50 x^{\frac{- 1 + 2}{2}}]$

Step 4.5

Add $- 1$ and $2$ .

$\frac{d}{d x} [50 x^{\frac{1}{2}}]$

Step 5

Since $50$ is constant with respect to $x$ , the derivative of $50 x^{\frac{1}{2}}$ with respect to $x$ is $50 \frac{d}{d x} [x^{\frac{1}{2}}]$ .

$50 \frac{d}{d x} [x^{\frac{1}{2}}]$

Step 6

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

$50 (\frac{1}{2} x^{\frac{1}{2} - 1})$

Step 7

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

$50 (\frac{1}{2} x^{\frac{1}{2} - 1 \cdot \frac{2}{2}})$

Step 8

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

$50 (\frac{1}{2} x^{\frac{1}{2} + \frac{- 1 \cdot 2}{2}})$

Step 9

Combine the numerators over the common denominator.

$50 (\frac{1}{2} x^{\frac{1 - 1 \cdot 2}{2}})$

Step 10

Simplify the numerator.

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Step 10.1

Multiply $- 1$ by $2$ .

$50 (\frac{1}{2} x^{\frac{1 - 2}{2}})$

Step 10.2

Subtract $2$ from $1$ .

$50 (\frac{1}{2} x^{\frac{- 1}{2}})$

Step 11

Move the negative in front of the fraction.

$50 (\frac{1}{2} x^{- \frac{1}{2}})$

Step 12

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

$50 \frac{x^{- \frac{1}{2}}}{2}$

Step 13

Combine $50$ and $\frac{x^{- \frac{1}{2}}}{2}$ .

$\frac{50 x^{- \frac{1}{2}}}{2}$

Step 14

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

$\frac{50}{2 x^{\frac{1}{2}}}$

Step 15

Factor $2$ out of $50$ .

$\frac{2 \cdot 25}{2 x^{\frac{1}{2}}}$

Step 16

Cancel the common factors.

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Step 16.1

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

$\frac{2 \cdot 25}{2 (x^{\frac{1}{2}})}$

Step 16.2

Cancel the common factor.

$\frac{2 \cdot 25}{2 x^{\frac{1}{2}}}$

Step 16.3

Rewrite the expression.

$\frac{25}{x^{\frac{1}{2}}}$