Calculus Examples

$y = \sqrt{5 x}$

Step 1

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

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

Step 2

Differentiate both sides of the equation.

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

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

Differentiate using the Constant Multiple Rule.

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Step 4.1.1

Apply the product rule to $5 x$ .

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

Step 4.1.2

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

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

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 $f (y) = y^{\frac{1}{2}}$ and $g (y) = x$ .

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Step 4.2.1

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

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

Step 4.2.2

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

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

Step 4.2.3

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

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

Step 4.3

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

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

Step 4.4

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

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

Step 4.5

Combine the numerators over the common denominator.

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

Step 4.6

Simplify the numerator.

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Step 4.6.1

Multiply $- 1$ by $2$ .

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

Step 4.6.2

Subtract $2$ from $1$ .

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

Step 4.7

Move the negative in front of the fraction.

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

Step 4.8

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

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

Step 4.9

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

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

Step 4.10

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

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

Step 4.11

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

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

Step 4.12

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

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

Step 5

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

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

Step 6

Solve for $x'$ .

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Step 6.1

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

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

Step 6.2

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

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

Step 6.3

Simplify.

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Step 6.3.1

Simplify the left side.

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Step 6.3.1.1

Simplify $\frac{5^{\frac{1}{2}} x'}{2 x^{\frac{1}{2}}} (2 x^{\frac{1}{2}})$ .

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Step 6.3.1.1.1

Rewrite using the commutative property of multiplication.

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

Step 6.3.1.1.2

Cancel the common factor of $2$ .

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Step 6.3.1.1.2.1

Cancel the common factor.

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

Step 6.3.1.1.2.2

Rewrite the expression.

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

Step 6.3.1.1.3

Cancel the common factor of $x^{\frac{1}{2}}$ .

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Step 6.3.1.1.3.1

Cancel the common factor.

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

Step 6.3.1.1.3.2

Rewrite the expression.

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

Step 6.3.1.1.4

Reorder factors in $5^{\frac{1}{2}} x'$ .

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

Step 6.3.2

Simplify the right side.

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Step 6.3.2.1

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

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

Step 6.4

Divide each term in $x' \cdot 5^{\frac{1}{2}} = 2 x^{\frac{1}{2}}$ by $5^{\frac{1}{2}}$ and simplify.

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Step 6.4.1

Divide each term in $x' \cdot 5^{\frac{1}{2}} = 2 x^{\frac{1}{2}}$ by $5^{\frac{1}{2}}$ .

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

Step 6.4.2

Simplify the left side.

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Step 6.4.2.1

Cancel the common factor.

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

Step 6.4.2.2

Divide $x'$ by $1$ .

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

Step 7

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

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