Calculus Examples

$y = \frac{1}{4 \sqrt{x}} \cdot (1 + \frac{1}{x \sqrt{x}})$

Step 1

Multiply $\frac{1}{4 \sqrt{x}}$ by $\frac{\sqrt{x}}{\sqrt{x}}$ .

$\frac{d}{d y} [\frac{1}{4 \sqrt{x}} \cdot \frac{\sqrt{x}}{\sqrt{x}} \cdot (1 + \frac{1}{x \sqrt{x}})]$

Step 2

Simplify terms.

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Step 2.1

Combine and simplify the denominator.

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Step 2.1.1

Multiply $\frac{1}{4 \sqrt{x}}$ by $\frac{\sqrt{x}}{\sqrt{x}}$ .

$\frac{d}{d y} [\frac{\sqrt{x}}{4 \sqrt{x} \sqrt{x}} \cdot (1 + \frac{1}{x \sqrt{x}})]$

Step 2.1.2

Move $\sqrt{x}$ .

$\frac{d}{d y} [\frac{\sqrt{x}}{4 (\sqrt{x} \sqrt{x})} \cdot (1 + \frac{1}{x \sqrt{x}})]$

Step 2.1.3

Raise $\sqrt{x}$ to the power of $1$ .

$\frac{d}{d y} [\frac{\sqrt{x}}{4 ({\sqrt{x}}^{1} \sqrt{x})} \cdot (1 + \frac{1}{x \sqrt{x}})]$

Step 2.1.4

Raise $\sqrt{x}$ to the power of $1$ .

$\frac{d}{d y} [\frac{\sqrt{x}}{4 ({\sqrt{x}}^{1} {\sqrt{x}}^{1})} \cdot (1 + \frac{1}{x \sqrt{x}})]$

Step 2.1.5

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

$\frac{d}{d y} [\frac{\sqrt{x}}{4 {\sqrt{x}}^{1 + 1}} \cdot (1 + \frac{1}{x \sqrt{x}})]$

Step 2.1.6

Add $1$ and $1$ .

$\frac{d}{d y} [\frac{\sqrt{x}}{4 {\sqrt{x}}^{2}} \cdot (1 + \frac{1}{x \sqrt{x}})]$

Step 2.1.7

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

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Step 2.1.7.1

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

$\frac{d}{d y} [\frac{\sqrt{x}}{4 {(x^{\frac{1}{2}})}^{2}} \cdot (1 + \frac{1}{x \sqrt{x}})]$

Step 2.1.7.2

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

$\frac{d}{d y} [\frac{\sqrt{x}}{4 x^{\frac{1}{2} \cdot 2}} \cdot (1 + \frac{1}{x \sqrt{x}})]$

Step 2.1.7.3

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

$\frac{d}{d y} [\frac{\sqrt{x}}{4 x^{\frac{2}{2}}} \cdot (1 + \frac{1}{x \sqrt{x}})]$

Step 2.1.7.4

Cancel the common factor of $2$ .

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Step 2.1.7.4.1

Cancel the common factor.

$\frac{d}{d y} [\frac{\sqrt{x}}{4 x^{\frac{2}{2}}} \cdot (1 + \frac{1}{x \sqrt{x}})]$

Step 2.1.7.4.2

Rewrite the expression.

$\frac{d}{d y} [\frac{\sqrt{x}}{4 x^{1}} \cdot (1 + \frac{1}{x \sqrt{x}})]$

Step 2.1.7.5

Simplify.

$\frac{d}{d y} [\frac{\sqrt{x}}{4 x} \cdot (1 + \frac{1}{x \sqrt{x}})]$

Step 2.2

Simplify each term.

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Step 2.2.1

Multiply $\frac{1}{x \sqrt{x}}$ by $\frac{\sqrt{x}}{\sqrt{x}}$ .

$\frac{d}{d y} [\frac{\sqrt{x}}{4 x} \cdot (1 + \frac{1}{x \sqrt{x}} \cdot \frac{\sqrt{x}}{\sqrt{x}})]$

Step 2.2.2

Combine and simplify the denominator.

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Step 2.2.2.1

Multiply $\frac{1}{x \sqrt{x}}$ by $\frac{\sqrt{x}}{\sqrt{x}}$ .

$\frac{d}{d y} [\frac{\sqrt{x}}{4 x} \cdot (1 + \frac{\sqrt{x}}{x \sqrt{x} \sqrt{x}})]$

Step 2.2.2.2

Move $\sqrt{x}$ .

$\frac{d}{d y} [\frac{\sqrt{x}}{4 x} \cdot (1 + \frac{\sqrt{x}}{x (\sqrt{x} \sqrt{x})})]$

Step 2.2.2.3

Raise $\sqrt{x}$ to the power of $1$ .

$\frac{d}{d y} [\frac{\sqrt{x}}{4 x} \cdot (1 + \frac{\sqrt{x}}{x ({\sqrt{x}}^{1} \sqrt{x})})]$

Step 2.2.2.4

Raise $\sqrt{x}$ to the power of $1$ .

$\frac{d}{d y} [\frac{\sqrt{x}}{4 x} \cdot (1 + \frac{\sqrt{x}}{x ({\sqrt{x}}^{1} {\sqrt{x}}^{1})})]$

Step 2.2.2.5

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

$\frac{d}{d y} [\frac{\sqrt{x}}{4 x} \cdot (1 + \frac{\sqrt{x}}{x {\sqrt{x}}^{1 + 1}})]$

Step 2.2.2.6

Add $1$ and $1$ .

$\frac{d}{d y} [\frac{\sqrt{x}}{4 x} \cdot (1 + \frac{\sqrt{x}}{x {\sqrt{x}}^{2}})]$

Step 2.2.2.7

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

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Step 2.2.2.7.1

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

$\frac{d}{d y} [\frac{\sqrt{x}}{4 x} \cdot (1 + \frac{\sqrt{x}}{x {(x^{\frac{1}{2}})}^{2}})]$

Step 2.2.2.7.2

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

$\frac{d}{d y} [\frac{\sqrt{x}}{4 x} \cdot (1 + \frac{\sqrt{x}}{x \cdot x^{\frac{1}{2} \cdot 2}})]$

Step 2.2.2.7.3

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

$\frac{d}{d y} [\frac{\sqrt{x}}{4 x} \cdot (1 + \frac{\sqrt{x}}{x \cdot x^{\frac{2}{2}}})]$

Step 2.2.2.7.4

Cancel the common factor of $2$ .

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Step 2.2.2.7.4.1

Cancel the common factor.

$\frac{d}{d y} [\frac{\sqrt{x}}{4 x} \cdot (1 + \frac{\sqrt{x}}{x \cdot x^{\frac{2}{2}}})]$

Step 2.2.2.7.4.2

Rewrite the expression.

$\frac{d}{d y} [\frac{\sqrt{x}}{4 x} \cdot (1 + \frac{\sqrt{x}}{x \cdot x^{1}})]$

Step 2.2.2.7.5

Simplify.

$\frac{d}{d y} [\frac{\sqrt{x}}{4 x} \cdot (1 + \frac{\sqrt{x}}{x \cdot x})]$

Step 2.2.3

Multiply $x$ by $x$ .

$\frac{d}{d y} [\frac{\sqrt{x}}{4 x} \cdot (1 + \frac{\sqrt{x}}{x^{2}})]$

Step 3

Since $\frac{\sqrt{x}}{4 x} (1 + \frac{\sqrt{x}}{x^{2}})$ is constant with respect to $y$ , the derivative of $\frac{\sqrt{x}}{4 x} (1 + \frac{\sqrt{x}}{x^{2}})$ with respect to $y$ is $0$ .

$0$