Algebra Examples

$y = 1 + \sqrt{5 x}$

Step 1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{5 x}$ as ${(5 x)}^{\frac{1}{2}}$ .

$y = 1 + {(5 x)}^{\frac{1}{2}}$

Step 2

Differentiate both sides of the equation.

$\frac{d}{d x} (y) = \frac{d}{d x} (1 + {(5 x)}^{\frac{1}{2}})$

Step 3

The derivative of $y$ with respect to $x$ is $y'$ .

$y'$

Step 4

Differentiate the right side of the equation.

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Step 4.1

Differentiate.

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Step 4.1.1

By the Sum Rule, the derivative of $1 + {(5 x)}^{\frac{1}{2}}$ with respect to $x$ is $\frac{d}{d x} [1] + \frac{d}{d x} [{(5 x)}^{\frac{1}{2}}]$ .

$\frac{d}{d x} [1] + \frac{d}{d x} [{(5 x)}^{\frac{1}{2}}]$

Step 4.1.2

Since $1$ is constant with respect to $x$ , the derivative of $1$ with respect to $x$ is $0$ .

$0 + \frac{d}{d x} [{(5 x)}^{\frac{1}{2}}]$

Step 4.2

Evaluate $\frac{d}{d x} [{(5 x)}^{\frac{1}{2}}]$ .

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Step 4.2.1

Factor $5$ out of $5 x$ .

$0 + \frac{d}{d x} [{(5 (x))}^{\frac{1}{2}}]$

Step 4.2.2

Apply the product rule to $5 (x)$ .

$0 + \frac{d}{d x} [5^{\frac{1}{2}} x^{\frac{1}{2}}]$

Step 4.2.3

Since $5^{\frac{1}{2}}$ is constant with respect to $x$ , the derivative of $5^{\frac{1}{2}} x^{\frac{1}{2}}$ with respect to $x$ is $5^{\frac{1}{2}} \frac{d}{d x} [x^{\frac{1}{2}}]$ .

$0 + 5^{\frac{1}{2}} \frac{d}{d x} [x^{\frac{1}{2}}]$

Step 4.2.4

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

$0 + 5^{\frac{1}{2}} (\frac{1}{2} x^{\frac{1}{2} - 1})$

Step 4.2.5

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

$0 + 5^{\frac{1}{2}} (\frac{1}{2} x^{\frac{1}{2} - 1 \cdot \frac{2}{2}})$

Step 4.2.6

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

$0 + 5^{\frac{1}{2}} (\frac{1}{2} x^{\frac{1}{2} + \frac{- 1 \cdot 2}{2}})$

Step 4.2.7

Combine the numerators over the common denominator.

$0 + 5^{\frac{1}{2}} (\frac{1}{2} x^{\frac{1 - 1 \cdot 2}{2}})$

Step 4.2.8

Simplify the numerator.

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Step 4.2.8.1

Multiply $- 1$ by $2$ .

$0 + 5^{\frac{1}{2}} (\frac{1}{2} x^{\frac{1 - 2}{2}})$

Step 4.2.8.2

Subtract $2$ from $1$ .

$0 + 5^{\frac{1}{2}} (\frac{1}{2} x^{\frac{- 1}{2}})$

Step 4.2.9

Move the negative in front of the fraction.

$0 + 5^{\frac{1}{2}} (\frac{1}{2} x^{- \frac{1}{2}})$

Step 4.2.10

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

$0 + 5^{\frac{1}{2}} \frac{x^{- \frac{1}{2}}}{2}$

Step 4.2.11

Combine $5^{\frac{1}{2}}$ and $\frac{x^{- \frac{1}{2}}}{2}$ .

$0 + \frac{5^{\frac{1}{2}} x^{- \frac{1}{2}}}{2}$

Step 4.2.12

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

$0 + \frac{5^{\frac{1}{2}}}{2 x^{\frac{1}{2}}}$

Step 4.3

Add $0$ and $\frac{5^{\frac{1}{2}}}{2 x^{\frac{1}{2}}}$ .

$\frac{5^{\frac{1}{2}}}{2 x^{\frac{1}{2}}}$

Step 5

Reform the equation by setting the left side equal to the right side.

$y' = \frac{5^{\frac{1}{2}}}{2 x^{\frac{1}{2}}}$

Step 6

Replace $y'$ with $\frac{d y}{d x}$ .

$\frac{d y}{d x} = \frac{5^{\frac{1}{2}}}{2 x^{\frac{1}{2}}}$