Algebra Examples

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

Step 1

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

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

Step 2

Differentiate both sides of the equation.

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

Step 3

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

$1$

Step 4

Differentiate the right side of the equation.

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Step 4.1

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

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Step 4.1.1

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

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

Step 4.1.2

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

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

Step 4.1.3

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

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

Step 4.1.4

Combine the numerators over the common denominator.

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

Step 4.1.5

Simplify the numerator.

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Step 4.1.5.1

Multiply $2$ by $2$ .

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

Step 4.1.5.2

Add $4$ and $1$ .

$\frac{d}{d y} [x^{\frac{5}{2}}]$

Step 4.2

Differentiate using the chain rule, which states that $\frac{d}{d y} [f (g (y))]$ is $f' (g (y)) g' (y)$ where $f (y) = y^{\frac{5}{2}}$ and $g (y) = x$ .

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Step 4.2.1

To apply the Chain Rule, set $u$ as $x$ .

$\frac{d}{d u} [u^{\frac{5}{2}}] \frac{d}{d y} [x]$

Step 4.2.2

Differentiate using the Power Rule which states that $\frac{d}{d u} [u^{n}]$ is $n u^{n - 1}$ where $n = \frac{5}{2}$ .

$\frac{5}{2} u^{\frac{5}{2} - 1} \frac{d}{d y} [x]$

Step 4.2.3

Replace all occurrences of $u$ with $x$ .

$\frac{5}{2} x^{\frac{5}{2} - 1} \frac{d}{d y} [x]$

Step 4.3

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

$\frac{5}{2} x^{\frac{5}{2} - 1 \cdot \frac{2}{2}} \frac{d}{d y} [x]$

Step 4.4

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

$\frac{5}{2} x^{\frac{5}{2} + \frac{- 1 \cdot 2}{2}} \frac{d}{d y} [x]$

Step 4.5

Combine the numerators over the common denominator.

$\frac{5}{2} x^{\frac{5 - 1 \cdot 2}{2}} \frac{d}{d y} [x]$

Step 4.6

Simplify the numerator.

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Step 4.6.1

Multiply $- 1$ by $2$ .

$\frac{5}{2} x^{\frac{5 - 2}{2}} \frac{d}{d y} [x]$

Step 4.6.2

Subtract $2$ from $5$ .

$\frac{5}{2} x^{\frac{3}{2}} \frac{d}{d y} [x]$

Step 4.7

Combine $\frac{5}{2}$ and $x^{\frac{3}{2}}$ .

$\frac{5 x^{\frac{3}{2}}}{2} \frac{d}{d y} [x]$

Step 4.8

Rewrite $\frac{d}{d y} [x]$ as $x'$ .

$\frac{5 x^{\frac{3}{2}}}{2} x'$

Step 4.9

Combine $\frac{5 x^{\frac{3}{2}}}{2}$ and $x'$ .

$\frac{5 x^{\frac{3}{2}} x'}{2}$

Step 5

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

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

Step 6

Solve for $x'$ .

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Step 6.1

Rewrite the equation as $\frac{5 x^{\frac{3}{2}} x'}{2} = 1$ .

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

Step 6.2

Multiply both sides of the equation by $\frac{2}{5}$ .

$\frac{2}{5} \cdot \frac{5 x^{\frac{3}{2}} x'}{2} = \frac{2}{5} \cdot 1$

Step 6.3

Simplify both sides of the equation.

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Step 6.3.1

Simplify the left side.

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Step 6.3.1.1

Simplify $\frac{2}{5} \cdot \frac{5 x^{\frac{3}{2}} x'}{2}$ .

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Step 6.3.1.1.1

Combine.

$\frac{2 (5 x^{\frac{3}{2}} x')}{5 \cdot 2} = \frac{2}{5} \cdot 1$

Step 6.3.1.1.2

Cancel the common factor.

$\frac{2 (5 x^{\frac{3}{2}} x')}{5 \cdot 2} = \frac{2}{5} \cdot 1$

Step 6.3.1.1.3

Rewrite the expression.

$\frac{5 x^{\frac{3}{2}} x'}{5} = \frac{2}{5} \cdot 1$

Step 6.3.1.1.4

Cancel the common factor.

$\frac{5 x^{\frac{3}{2}} x'}{5} = \frac{2}{5} \cdot 1$

Step 6.3.1.1.5

Simplify the expression.

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Step 6.3.1.1.5.1

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

$x^{\frac{3}{2}} x' = \frac{2}{5} \cdot 1$

Step 6.3.1.1.5.2

Reorder factors in $x^{\frac{3}{2}} x'$ .

$x' x^{\frac{3}{2}} = \frac{2}{5} \cdot 1$

Step 6.3.2

Simplify the right side.

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Step 6.3.2.1

Multiply $\frac{2}{5}$ by $1$ .

$x' x^{\frac{3}{2}} = \frac{2}{5}$

Step 6.4

Divide each term in $x' x^{\frac{3}{2}} = \frac{2}{5}$ by $x^{\frac{3}{2}}$ and simplify.

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Step 6.4.1

Divide each term in $x' x^{\frac{3}{2}} = \frac{2}{5}$ by $x^{\frac{3}{2}}$ .

$\frac{x' x^{\frac{3}{2}}}{x^{\frac{3}{2}}} = \frac{\frac{2}{5}}{x^{\frac{3}{2}}}$

Step 6.4.2

Simplify the left side.

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Step 6.4.2.1

Cancel the common factor.

$\frac{x' x^{\frac{3}{2}}}{x^{\frac{3}{2}}} = \frac{\frac{2}{5}}{x^{\frac{3}{2}}}$

Step 6.4.2.2

Divide $x'$ by $1$ .

$x' = \frac{\frac{2}{5}}{x^{\frac{3}{2}}}$

Step 6.4.3

Simplify the right side.

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Step 6.4.3.1

Multiply the numerator by the reciprocal of the denominator.

$x' = \frac{2}{5} \cdot \frac{1}{x^{\frac{3}{2}}}$

Step 6.4.3.2

Combine.

$x' = \frac{2 \cdot 1}{5 x^{\frac{3}{2}}}$

Step 6.4.3.3

Multiply $2$ by $1$ .

$x' = \frac{2}{5 x^{\frac{3}{2}}}$

Step 7

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

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