Calculus Examples

$y = \frac{1}{\sqrt{x}}$

Step 1

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

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

Step 2

Differentiate both sides of the equation.

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

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

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

Step 4.2

Simplify the expression.

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Step 4.2.1

Multiply $- 1$ by $1$ .

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

Step 4.2.2

Multiply the exponents in ${(x^{\frac{1}{2}})}^{2}$ .

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Step 4.2.2.1

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

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

Step 4.2.2.2

Cancel the common factor of $2$ .

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Step 4.2.2.2.1

Cancel the common factor.

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

Step 4.2.2.2.2

Rewrite the expression.

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

Step 4.3

Simplify.

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

Step 4.4

Differentiate using the Constant Rule.

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Step 4.4.1

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

$\frac{x^{\frac{1}{2}} \cdot 0 - \frac{d}{d y} [x^{\frac{1}{2}}]}{x}$

Step 4.4.2

Simplify the expression.

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Step 4.4.2.1

Multiply $x^{\frac{1}{2}}$ by $0$ .

$\frac{0 - \frac{d}{d y} [x^{\frac{1}{2}}]}{x}$

Step 4.4.2.2

Subtract $\frac{d}{d y} [x^{\frac{1}{2}}]$ from $0$ .

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

Step 4.4.2.3

Move the negative in front of the fraction.

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

Step 4.5

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 4.5.1

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

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

Step 4.5.2

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

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

Step 4.5.3

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

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

Step 4.6

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

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

Step 4.7

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

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

Step 4.8

Combine the numerators over the common denominator.

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

Step 4.9

Simplify the numerator.

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Step 4.9.1

Multiply $- 1$ by $2$ .

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

Step 4.9.2

Subtract $2$ from $1$ .

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

Step 4.10

Move the negative in front of the fraction.

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

Step 4.11

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

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

Step 4.12

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

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

Step 4.13

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

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

Step 4.14

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

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

Step 4.15

Rewrite $\frac{\frac{x'}{2 x^{\frac{1}{2}}}}{x}$ as a product.

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

Step 4.16

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

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

Step 4.17

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

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

Step 4.18

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

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

Step 4.19

Write $1$ as a fraction with a common denominator.

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

Step 4.20

Combine the numerators over the common denominator.

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

Step 4.21

Add $2$ and $1$ .

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

Step 5

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

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

Step 6

Solve for $x'$ .

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Step 6.1

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

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

Step 6.2

Divide each term in $- \frac{x'}{2 x^{\frac{3}{2}}} = 1$ by $- 1$ and simplify.

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Step 6.2.1

Divide each term in $- \frac{x'}{2 x^{\frac{3}{2}}} = 1$ by $- 1$ .

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

Step 6.2.2

Simplify the left side.

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Step 6.2.2.1

Dividing two negative values results in a positive value.

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

Step 6.2.2.2

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

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

Step 6.2.3

Simplify the right side.

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Step 6.2.3.1

Divide $1$ by $- 1$ .

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

Step 6.3

Multiply both sides by $2 x^{\frac{3}{2}}$ .

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

Step 6.4

Simplify.

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Step 6.4.1

Simplify the left side.

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Step 6.4.1.1

Simplify $\frac{x'}{2 x^{\frac{3}{2}}} (2 x^{\frac{3}{2}})$ .

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Step 6.4.1.1.1

Rewrite using the commutative property of multiplication.

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

Step 6.4.1.1.2

Cancel the common factor of $2$ .

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Step 6.4.1.1.2.1

Cancel the common factor.

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

Step 6.4.1.1.2.2

Rewrite the expression.

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

Step 6.4.1.1.3

Cancel the common factor of $x^{\frac{3}{2}}$ .

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Step 6.4.1.1.3.1

Cancel the common factor.

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

Step 6.4.1.1.3.2

Rewrite the expression.

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

Step 6.4.2

Simplify the right side.

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Step 6.4.2.1

Multiply $2$ by $- 1$ .

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

Step 7

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

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