Calculus Examples

$\sqrt{50 x} - \sqrt{18 x}$

Step 1

By the Sum Rule, the derivative of $\sqrt{50 x} - \sqrt{18 x}$ with respect to $x$ is $\frac{d}{d x} [\sqrt{50 x}] + \frac{d}{d x} [- \sqrt{18 x}]$ .

$\frac{d}{d x} [\sqrt{50 x}] + \frac{d}{d x} [- \sqrt{18 x}]$

Step 2

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

Tap for more steps...

Step 2.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{50 x}$ as ${(50 x)}^{\frac{1}{2}}$ .

$\frac{d}{d x} [{(50 x)}^{\frac{1}{2}}] + \frac{d}{d x} [- \sqrt{18 x}]$

Step 2.2

Factor $50$ out of $50 x$ .

$\frac{d}{d x} [{(50 (x))}^{\frac{1}{2}}] + \frac{d}{d x} [- \sqrt{18 x}]$

Step 2.3

Apply the product rule to $50 (x)$ .

$\frac{d}{d x} [50^{\frac{1}{2}} x^{\frac{1}{2}}] + \frac{d}{d x} [- \sqrt{18 x}]$

Step 2.4

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

$50^{\frac{1}{2}} \frac{d}{d x} [x^{\frac{1}{2}}] + \frac{d}{d x} [- \sqrt{18 x}]$

Step 2.5

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

$50^{\frac{1}{2}} (\frac{1}{2} x^{\frac{1}{2} - 1}) + \frac{d}{d x} [- \sqrt{18 x}]$

Step 2.6

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

$50^{\frac{1}{2}} (\frac{1}{2} x^{\frac{1}{2} - 1 \cdot \frac{2}{2}}) + \frac{d}{d x} [- \sqrt{18 x}]$

Step 2.7

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

$50^{\frac{1}{2}} (\frac{1}{2} x^{\frac{1}{2} + \frac{- 1 \cdot 2}{2}}) + \frac{d}{d x} [- \sqrt{18 x}]$

Step 2.8

Combine the numerators over the common denominator.

$50^{\frac{1}{2}} (\frac{1}{2} x^{\frac{1 - 1 \cdot 2}{2}}) + \frac{d}{d x} [- \sqrt{18 x}]$

Step 2.9

Simplify the numerator.

Tap for more steps...

Step 2.9.1

Multiply $- 1$ by $2$ .

$50^{\frac{1}{2}} (\frac{1}{2} x^{\frac{1 - 2}{2}}) + \frac{d}{d x} [- \sqrt{18 x}]$

Step 2.9.2

Subtract $2$ from $1$ .

$50^{\frac{1}{2}} (\frac{1}{2} x^{\frac{- 1}{2}}) + \frac{d}{d x} [- \sqrt{18 x}]$

Step 2.10

Move the negative in front of the fraction.

$50^{\frac{1}{2}} (\frac{1}{2} x^{- \frac{1}{2}}) + \frac{d}{d x} [- \sqrt{18 x}]$

Step 2.11

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

$50^{\frac{1}{2}} \frac{x^{- \frac{1}{2}}}{2} + \frac{d}{d x} [- \sqrt{18 x}]$

Step 2.12

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

$\frac{50^{\frac{1}{2}} x^{- \frac{1}{2}}}{2} + \frac{d}{d x} [- \sqrt{18 x}]$

Step 2.13

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

$\frac{50^{\frac{1}{2}}}{2 x^{\frac{1}{2}}} + \frac{d}{d x} [- \sqrt{18 x}]$

Step 3

Evaluate $\frac{d}{d x} [- \sqrt{18 x}]$ .

Tap for more steps...

Step 3.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{18 x}$ as ${(18 x)}^{\frac{1}{2}}$ .

$\frac{50^{\frac{1}{2}}}{2 x^{\frac{1}{2}}} + \frac{d}{d x} [- {(18 x)}^{\frac{1}{2}}]$

Step 3.2

Factor $18$ out of $18 x$ .

$\frac{50^{\frac{1}{2}}}{2 x^{\frac{1}{2}}} + \frac{d}{d x} [- {(18 (x))}^{\frac{1}{2}}]$

Step 3.3

Apply the product rule to $18 (x)$ .

$\frac{50^{\frac{1}{2}}}{2 x^{\frac{1}{2}}} + \frac{d}{d x} [- (18^{\frac{1}{2}} x^{\frac{1}{2}})]$

Step 3.4

Since $- 18^{\frac{1}{2}}$ is constant with respect to $x$ , the derivative of $- 18^{\frac{1}{2}} x^{\frac{1}{2}}$ with respect to $x$ is $- 18^{\frac{1}{2}} \frac{d}{d x} [x^{\frac{1}{2}}]$ .

$\frac{50^{\frac{1}{2}}}{2 x^{\frac{1}{2}}} - 18^{\frac{1}{2}} \frac{d}{d x} [x^{\frac{1}{2}}]$

Step 3.5

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

$\frac{50^{\frac{1}{2}}}{2 x^{\frac{1}{2}}} - 18^{\frac{1}{2}} (\frac{1}{2} x^{\frac{1}{2} - 1})$

Step 3.6

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

$\frac{50^{\frac{1}{2}}}{2 x^{\frac{1}{2}}} - 18^{\frac{1}{2}} (\frac{1}{2} x^{\frac{1}{2} - 1 \cdot \frac{2}{2}})$

Step 3.7

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

$\frac{50^{\frac{1}{2}}}{2 x^{\frac{1}{2}}} - 18^{\frac{1}{2}} (\frac{1}{2} x^{\frac{1}{2} + \frac{- 1 \cdot 2}{2}})$

Step 3.8

Combine the numerators over the common denominator.

$\frac{50^{\frac{1}{2}}}{2 x^{\frac{1}{2}}} - 18^{\frac{1}{2}} (\frac{1}{2} x^{\frac{1 - 1 \cdot 2}{2}})$

Step 3.9

Simplify the numerator.

Tap for more steps...

Step 3.9.1

Multiply $- 1$ by $2$ .

$\frac{50^{\frac{1}{2}}}{2 x^{\frac{1}{2}}} - 18^{\frac{1}{2}} (\frac{1}{2} x^{\frac{1 - 2}{2}})$

Step 3.9.2

Subtract $2$ from $1$ .

$\frac{50^{\frac{1}{2}}}{2 x^{\frac{1}{2}}} - 18^{\frac{1}{2}} (\frac{1}{2} x^{\frac{- 1}{2}})$

Step 3.10

Move the negative in front of the fraction.

$\frac{50^{\frac{1}{2}}}{2 x^{\frac{1}{2}}} - 18^{\frac{1}{2}} (\frac{1}{2} x^{- \frac{1}{2}})$

Step 3.11

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

$\frac{50^{\frac{1}{2}}}{2 x^{\frac{1}{2}}} - 18^{\frac{1}{2}} \frac{x^{- \frac{1}{2}}}{2}$

Step 3.12

Combine $\frac{x^{- \frac{1}{2}}}{2}$ and $18^{\frac{1}{2}}$ .

$\frac{50^{\frac{1}{2}}}{2 x^{\frac{1}{2}}} - \frac{x^{- \frac{1}{2}} \cdot 18^{\frac{1}{2}}}{2}$

Step 3.13

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

$\frac{50^{\frac{1}{2}}}{2 x^{\frac{1}{2}}} - \frac{18^{\frac{1}{2}}}{2 x^{\frac{1}{2}}}$