Calculus Examples

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

Step 1

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 $f (x) = 1$ and $g (x) = \sqrt{x}$ .

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

Step 2

Differentiate.

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Step 2.1

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

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

Step 2.2

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

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

Step 2.3

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

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

Step 3

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

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

Step 4

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

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

Step 5

Combine the numerators over the common denominator.

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

Step 6

Simplify the numerator.

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Step 6.1

Multiply $- 1$ by $2$ .

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

Step 6.2

Subtract $2$ from $1$ .

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

Step 7

Move the negative in front of the fraction.

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

Step 8

Simplify.

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Step 8.1

Rewrite the expression using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

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

Step 8.2

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

$\frac{\sqrt{x} \cdot 0 - 1 \cdot 1 \frac{1}{2 x^{\frac{1}{2}}}}{{\sqrt{x}}^{2}}$

Step 9

Simplify.

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Step 9.1

Simplify the numerator.

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Step 9.1.1

Simplify each term.

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Step 9.1.1.1

Multiply $\sqrt{x}$ by $0$ .

$\frac{0 - 1 \cdot 1 \frac{1}{2 x^{\frac{1}{2}}}}{{\sqrt{x}}^{2}}$

Step 9.1.1.2

Multiply $- 1$ by $1$ .

$\frac{0 - \frac{1}{2 x^{\frac{1}{2}}}}{{\sqrt{x}}^{2}}$

Step 9.1.2

Subtract $\frac{1}{2 x^{\frac{1}{2}}}$ from $0$ .

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

Step 9.2

Combine terms.

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Step 9.2.1

Rewrite ${\sqrt{x}}^{2}$ as $x$ .

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Step 9.2.1.1

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

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

Step 9.2.1.2

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

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

Step 9.2.1.3

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

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

Step 9.2.1.4

Cancel the common factor of $2$ .

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Step 9.2.1.4.1

Cancel the common factor.

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

Step 9.2.1.4.2

Rewrite the expression.

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

Step 9.2.1.5

Simplify.

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

Step 9.2.2

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

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

Step 9.2.3

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

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

Step 9.2.4

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

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

Step 9.2.5

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

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

Step 9.2.6

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

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

Step 9.2.7

Combine the numerators over the common denominator.

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

Step 9.2.8

Add $2$ and $1$ .

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