Algebra Examples

$y = \frac{\sqrt{x} + x}{x^{2}}$

Step 1

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

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

Step 2

Differentiate both sides of the equation.

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

Step 3

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

$y'$

Step 4

Differentiate the right side of the equation.

Tap for more steps...

Step 4.1

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

Tap for more steps...

Step 4.1.1

Multiply by $1$ .

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

Step 4.1.2

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

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

Step 4.1.3

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

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

Step 4.1.4

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

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

Step 4.2

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

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

Step 4.3

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

Tap for more steps...

Step 4.3.1

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

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

Step 4.3.2

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

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

Step 4.3.3

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

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

Step 4.3.4

Combine the numerators over the common denominator.

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

Step 4.3.5

Simplify the numerator.

Tap for more steps...

Step 4.3.5.1

Multiply $2$ by $2$ .

$\frac{d}{d x} [\frac{1 + x^{\frac{1}{2}}}{x^{\frac{4 - 1}{2}}}]$

Step 4.3.5.2

Subtract $1$ from $4$ .

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

Step 4.4

Differentiate using the Quotient Rule which states that $\frac{d}{d x} [\frac{f (x)}{g (x)}]$ is $\frac{g (x) \frac{d}{d x} [f (x)] - f (x) \frac{d}{d x} [g (x)]}{{g (x)}^{2}}$ where $f (x) = 1 + x^{\frac{1}{2}}$ and $g (x) = x^{\frac{3}{2}}$ .

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

Step 4.5

Differentiate.

Tap for more steps...

Step 4.5.1

Multiply the exponents in ${(x^{\frac{3}{2}})}^{2}$ .

Tap for more steps...

Step 4.5.1.1

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

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

Step 4.5.1.2

Cancel the common factor of $2$ .

Tap for more steps...

Step 4.5.1.2.1

Cancel the common factor.

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

Step 4.5.1.2.2

Rewrite the expression.

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

Step 4.5.2

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

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

Step 4.5.3

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

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

Step 4.5.4

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

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

Step 4.5.5

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

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

Step 4.6

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

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

Step 4.7

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

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

Step 4.8

Combine the numerators over the common denominator.

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

Step 4.9

Simplify the numerator.

Tap for more steps...

Step 4.9.1

Multiply $- 1$ by $2$ .

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

Step 4.9.2

Subtract $2$ from $1$ .

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

Step 4.10

Move the negative in front of the fraction.

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

Step 4.11

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

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

Step 4.12

Combine $x^{\frac{3}{2}}$ and $\frac{x^{- \frac{1}{2}}}{2}$ .

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

Step 4.13

Multiply $x^{\frac{3}{2}}$ by $x^{- \frac{1}{2}}$ by adding the exponents.

Tap for more steps...

Step 4.13.1

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

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

Step 4.13.2

Combine the numerators over the common denominator.

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

Step 4.13.3

Subtract $1$ from $3$ .

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

Step 4.13.4

Divide $2$ by $2$ .

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

Step 4.14

Simplify $x^{1}$ .

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

Step 4.15

Multiply $\frac{\frac{x}{2} - (1 + x^{\frac{1}{2}}) \frac{d}{d x} [x^{\frac{3}{2}}]}{x^{3}}$ by $\frac{2}{2}$ .

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

Step 4.16

Simplify terms.

Tap for more steps...

Step 4.16.1

Combine.

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

Step 4.16.2

Apply the distributive property.

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

Step 4.16.3

Cancel the common factor of $2$ .

Tap for more steps...

Step 4.16.3.1

Cancel the common factor.

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

Step 4.16.3.2

Rewrite the expression.

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

Step 4.16.4

Multiply $- 1$ by $2$ .

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

Step 4.17

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = \frac{3}{2}$ .

$\frac{x - 2 (1 + x^{\frac{1}{2}}) (\frac{3}{2} x^{\frac{3}{2} - 1})}{2 x^{3}}$

Step 4.18

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

$\frac{x - 2 (1 + x^{\frac{1}{2}}) (\frac{3}{2} x^{\frac{3}{2} - 1 \cdot \frac{2}{2}})}{2 x^{3}}$

Step 4.19

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

$\frac{x - 2 (1 + x^{\frac{1}{2}}) (\frac{3}{2} x^{\frac{3}{2} + \frac{- 1 \cdot 2}{2}})}{2 x^{3}}$

Step 4.20

Combine the numerators over the common denominator.

$\frac{x - 2 (1 + x^{\frac{1}{2}}) (\frac{3}{2} x^{\frac{3 - 1 \cdot 2}{2}})}{2 x^{3}}$

Step 4.21

Simplify the numerator.

Tap for more steps...

Step 4.21.1

Multiply $- 1$ by $2$ .

$\frac{x - 2 (1 + x^{\frac{1}{2}}) (\frac{3}{2} x^{\frac{3 - 2}{2}})}{2 x^{3}}$

Step 4.21.2

Subtract $2$ from $3$ .

$\frac{x - 2 (1 + x^{\frac{1}{2}}) (\frac{3}{2} x^{\frac{1}{2}})}{2 x^{3}}$

Step 4.22

Combine $\frac{3}{2}$ and $x^{\frac{1}{2}}$ .

$\frac{x - 2 (1 + x^{\frac{1}{2}}) \frac{3 x^{\frac{1}{2}}}{2}}{2 x^{3}}$

Step 4.23

Combine $\frac{3 x^{\frac{1}{2}}}{2}$ and $- 2$ .

$\frac{x + \frac{3 x^{\frac{1}{2}} \cdot - 2}{2} (1 + x^{\frac{1}{2}})}{2 x^{3}}$

Step 4.24

Multiply $- 2$ by $3$ .

$\frac{x + \frac{- 6 x^{\frac{1}{2}}}{2} (1 + x^{\frac{1}{2}})}{2 x^{3}}$

Step 4.25

Factor $2$ out of $- 6 x^{\frac{1}{2}}$ .

$\frac{x + \frac{2 (- 3 x^{\frac{1}{2}})}{2} (1 + x^{\frac{1}{2}})}{2 x^{3}}$

Step 4.26

Cancel the common factors.

Tap for more steps...

Step 4.26.1

Factor $2$ out of $2$ .

$\frac{x + \frac{2 (- 3 x^{\frac{1}{2}})}{2 (1)} (1 + x^{\frac{1}{2}})}{2 x^{3}}$

Step 4.26.2

Cancel the common factor.

$\frac{x + \frac{2 (- 3 x^{\frac{1}{2}})}{2 \cdot 1} (1 + x^{\frac{1}{2}})}{2 x^{3}}$

Step 4.26.3

Rewrite the expression.

$\frac{x + \frac{- 3 x^{\frac{1}{2}}}{1} (1 + x^{\frac{1}{2}})}{2 x^{3}}$

Step 4.26.4

Divide $- 3 x^{\frac{1}{2}}$ by $1$ .

$\frac{x - 3 x^{\frac{1}{2}} (1 + x^{\frac{1}{2}})}{2 x^{3}}$

Step 4.27

Simplify.

Tap for more steps...

Step 4.27.1

Apply the distributive property.

$\frac{x - 3 x^{\frac{1}{2}} \cdot 1 - 3 x^{\frac{1}{2}} x^{\frac{1}{2}}}{2 x^{3}}$

Step 4.27.2

Simplify the numerator.

Tap for more steps...

Step 4.27.2.1

Simplify each term.

Tap for more steps...

Step 4.27.2.1.1

Multiply $- 3$ by $1$ .

$\frac{x - 3 x^{\frac{1}{2}} - 3 x^{\frac{1}{2}} x^{\frac{1}{2}}}{2 x^{3}}$

Step 4.27.2.1.2

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

Tap for more steps...

Step 4.27.2.1.2.1

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

$\frac{x - 3 x^{\frac{1}{2}} - 3 (x^{\frac{1}{2}} x^{\frac{1}{2}})}{2 x^{3}}$

Step 4.27.2.1.2.2

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

$\frac{x - 3 x^{\frac{1}{2}} - 3 x^{\frac{1}{2} + \frac{1}{2}}}{2 x^{3}}$

Step 4.27.2.1.2.3

Combine the numerators over the common denominator.

$\frac{x - 3 x^{\frac{1}{2}} - 3 x^{\frac{1 + 1}{2}}}{2 x^{3}}$

Step 4.27.2.1.2.4

Add $1$ and $1$ .

$\frac{x - 3 x^{\frac{1}{2}} - 3 x^{\frac{2}{2}}}{2 x^{3}}$

Step 4.27.2.1.2.5

Divide $2$ by $2$ .

$\frac{x - 3 x^{\frac{1}{2}} - 3 x^{1}}{2 x^{3}}$

Step 4.27.2.1.3

Simplify $- 3 x^{1}$ .

$\frac{x - 3 x^{\frac{1}{2}} - 3 x}{2 x^{3}}$

Step 4.27.2.2

Subtract $3 x$ from $x$ .

$\frac{- 2 x - 3 x^{\frac{1}{2}}}{2 x^{3}}$

Step 4.27.3

Simplify the numerator.

Tap for more steps...

Step 4.27.3.1

Factor $x^{\frac{1}{2}}$ out of $- 2 x - 3 x^{\frac{1}{2}}$ .

Tap for more steps...

Step 4.27.3.1.1

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

$\frac{x^{\frac{1}{2}} (- 2 x^{\frac{1}{2}}) - 3 x^{\frac{1}{2}}}{2 x^{3}}$

Step 4.27.3.1.2

Factor $x^{\frac{1}{2}}$ out of $- 3 x^{\frac{1}{2}}$ .

$\frac{x^{\frac{1}{2}} (- 2 x^{\frac{1}{2}}) + x^{\frac{1}{2}} \cdot - 3}{2 x^{3}}$

Step 4.27.3.1.3

Factor $x^{\frac{1}{2}}$ out of $x^{\frac{1}{2}} (- 2 x^{\frac{1}{2}}) + x^{\frac{1}{2}} \cdot - 3$ .

$\frac{x^{\frac{1}{2}} (- 2 x^{\frac{1}{2}} - 3)}{2 x^{3}}$

Step 4.27.3.2

Factor $- 1$ out of $- 2 x^{\frac{1}{2}} - 3$ .

Tap for more steps...

Step 4.27.3.2.1

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

$\frac{x^{\frac{1}{2}} (- (2 x^{\frac{1}{2}}) - 3)}{2 x^{3}}$

Step 4.27.3.2.2

Rewrite $- 3$ as $- 1 (3)$ .

$\frac{x^{\frac{1}{2}} (- (2 x^{\frac{1}{2}}) - 1 (3))}{2 x^{3}}$

Step 4.27.3.2.3

Factor $- 1$ out of $- (2 x^{\frac{1}{2}}) - 1 (3)$ .

$\frac{x^{\frac{1}{2}} (- (2 x^{\frac{1}{2}} + 3))}{2 x^{3}}$

$\frac{x^{\frac{1}{2}} \cdot - 1 (2 x^{\frac{1}{2}} + 3)}{2 x^{3}}$

Step 4.27.3.3

Factor out negative.

$\frac{- x^{\frac{1}{2}} (2 x^{\frac{1}{2}} + 3)}{2 x^{3}}$

Step 4.27.4

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

$\frac{- (2 x^{\frac{1}{2}} + 3)}{2 x^{3} x^{- \frac{1}{2}}}$

Step 4.27.5

Multiply $x^{3}$ by $x^{- \frac{1}{2}}$ by adding the exponents.

Tap for more steps...

Step 4.27.5.1

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

$\frac{- (2 x^{\frac{1}{2}} + 3)}{2 (x^{- \frac{1}{2}} x^{3})}$

Step 4.27.5.2

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

$\frac{- (2 x^{\frac{1}{2}} + 3)}{2 x^{- \frac{1}{2} + 3}}$

Step 4.27.5.3

To write $3$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$\frac{- (2 x^{\frac{1}{2}} + 3)}{2 x^{- \frac{1}{2} + 3 \cdot \frac{2}{2}}}$

Step 4.27.5.4

Combine $3$ and $\frac{2}{2}$ .

$\frac{- (2 x^{\frac{1}{2}} + 3)}{2 x^{- \frac{1}{2} + \frac{3 \cdot 2}{2}}}$

Step 4.27.5.5

Combine the numerators over the common denominator.

$\frac{- (2 x^{\frac{1}{2}} + 3)}{2 x^{\frac{- 1 + 3 \cdot 2}{2}}}$

Step 4.27.5.6

Simplify the numerator.

Tap for more steps...

Step 4.27.5.6.1

Multiply $3$ by $2$ .

$\frac{- (2 x^{\frac{1}{2}} + 3)}{2 x^{\frac{- 1 + 6}{2}}}$

Step 4.27.5.6.2

Add $- 1$ and $6$ .

$\frac{- (2 x^{\frac{1}{2}} + 3)}{2 x^{\frac{5}{2}}}$

Step 4.27.6

Move the negative in front of the fraction.

$- \frac{2 x^{\frac{1}{2}} + 3}{2 x^{\frac{5}{2}}}$

Step 5

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

$y' = - \frac{2 x^{\frac{1}{2}} + 3}{2 x^{\frac{5}{2}}}$

Step 6

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

$\frac{d y}{d x} = - \frac{2 x^{\frac{1}{2}} + 3}{2 x^{\frac{5}{2}}}$