Calculus Examples

$S = \frac{\sqrt{h w}}{60}$

Step 1

Differentiate using the Constant Multiple Rule.

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Step 1.1

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

$\frac{d}{d w} [\frac{{(h w)}^{\frac{1}{2}}}{60}]$

Step 1.2

Since $\frac{1}{60}$ is constant with respect to $w$ , the derivative of $\frac{{(h w)}^{\frac{1}{2}}}{60}$ with respect to $w$ is $\frac{1}{60} \frac{d}{d w} [{(h w)}^{\frac{1}{2}}]$ .

$\frac{1}{60} \frac{d}{d w} [{(h w)}^{\frac{1}{2}}]$

Step 2

Differentiate using the chain rule, which states that $\frac{d}{d w} [f (g (w))]$ is $f' (g (w)) g' (w)$ where $f (w) = w^{\frac{1}{2}}$ and $g (w) = h w$ .

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Step 2.1

To apply the Chain Rule, set $u$ as $h w$ .

$\frac{1}{60} (\frac{d}{d u} [u^{\frac{1}{2}}] \frac{d}{d w} [h w])$

Step 2.2

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

$\frac{1}{60} (\frac{1}{2} u^{\frac{1}{2} - 1} \frac{d}{d w} [h w])$

Step 2.3

Replace all occurrences of $u$ with $h w$ .

$\frac{1}{60} (\frac{1}{2} {(h w)}^{\frac{1}{2} - 1} \frac{d}{d w} [h w])$

Step 3

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

$\frac{1}{60} (\frac{1}{2} {(h w)}^{\frac{1}{2} - 1 \cdot \frac{2}{2}} \frac{d}{d w} [h w])$

Step 4

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

$\frac{1}{60} (\frac{1}{2} {(h w)}^{\frac{1}{2} + \frac{- 1 \cdot 2}{2}} \frac{d}{d w} [h w])$

Step 5

Combine the numerators over the common denominator.

$\frac{1}{60} (\frac{1}{2} {(h w)}^{\frac{1 - 1 \cdot 2}{2}} \frac{d}{d w} [h w])$

Step 6

Simplify the numerator.

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Step 6.1

Multiply $- 1$ by $2$ .

$\frac{1}{60} (\frac{1}{2} {(h w)}^{\frac{1 - 2}{2}} \frac{d}{d w} [h w])$

Step 6.2

Subtract $2$ from $1$ .

$\frac{1}{60} (\frac{1}{2} {(h w)}^{\frac{- 1}{2}} \frac{d}{d w} [h w])$

Step 7

Combine fractions.

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Step 7.1

Move the negative in front of the fraction.

$\frac{1}{60} (\frac{1}{2} {(h w)}^{- \frac{1}{2}} \frac{d}{d w} [h w])$

Step 7.2

Combine $\frac{1}{2}$ and ${(h w)}^{- \frac{1}{2}}$ .

$\frac{1}{60} (\frac{{(h w)}^{- \frac{1}{2}}}{2} \frac{d}{d w} [h w])$

Step 7.3

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

$\frac{1}{60} (\frac{1}{2 {(h w)}^{\frac{1}{2}}} \frac{d}{d w} [h w])$

Step 7.4

Multiply $\frac{1}{2 {(h w)}^{\frac{1}{2}}}$ by $\frac{1}{60}$ .

$\frac{1}{2 {(h w)}^{\frac{1}{2}} \cdot 60} \frac{d}{d w} [h w]$

Step 7.5

Multiply $60$ by $2$ .

$\frac{1}{120 {(h w)}^{\frac{1}{2}}} \frac{d}{d w} [h w]$

Step 8

Since $h$ is constant with respect to $w$ , the derivative of $h w$ with respect to $w$ is $h \frac{d}{d w} [w]$ .

$\frac{1}{120 {(h w)}^{\frac{1}{2}}} (h \frac{d}{d w} [w])$

Step 9

Combine $h$ and $\frac{1}{120 {(h w)}^{\frac{1}{2}}}$ .

$\frac{h}{120 {(h w)}^{\frac{1}{2}}} \frac{d}{d w} [w]$

Step 10

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

$\frac{h}{120 {(h w)}^{\frac{1}{2}}} \cdot 1$

Step 11

Multiply $\frac{h}{120 {(h w)}^{\frac{1}{2}}}$ by $1$ .

$\frac{h}{120 {(h w)}^{\frac{1}{2}}}$

Step 12

Simplify.

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Step 12.1

Apply the product rule to $h w$ .

$\frac{h}{120 (h^{\frac{1}{2}} w^{\frac{1}{2}})}$

Step 12.2

Combine terms.

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Step 12.2.1

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

$\frac{h \cdot h^{- \frac{1}{2}}}{120 w^{\frac{1}{2}}}$

Step 12.2.2

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

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Step 12.2.2.1

Multiply $h$ by $h^{- \frac{1}{2}}$ .

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Step 12.2.2.1.1

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

$\frac{h^{1} h^{- \frac{1}{2}}}{120 w^{\frac{1}{2}}}$

Step 12.2.2.1.2

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

$\frac{h^{1 - \frac{1}{2}}}{120 w^{\frac{1}{2}}}$

Step 12.2.2.2

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

$\frac{h^{\frac{2}{2} - \frac{1}{2}}}{120 w^{\frac{1}{2}}}$

Step 12.2.2.3

Combine the numerators over the common denominator.

$\frac{h^{\frac{2 - 1}{2}}}{120 w^{\frac{1}{2}}}$

Step 12.2.2.4

Subtract $1$ from $2$ .

$\frac{h^{\frac{1}{2}}}{120 w^{\frac{1}{2}}}$