Calculus Examples

$x = \sqrt{125 - p^{2}}$

Step 1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{125 - p^{2}}$ as ${(125 - p^{2})}^{\frac{1}{2}}$ .

$x = {(125 - p^{2})}^{\frac{1}{2}}$

Step 2

Differentiate both sides of the equation.

$\frac{d}{d p} (x) = \frac{d}{d p} ({(125 - p^{2})}^{\frac{1}{2}})$

Step 3

The derivative of $x$ with respect to $p$ is $x'$ .

$x'$

Step 4

Differentiate the right side of the equation.

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Step 4.1

Differentiate using the chain rule, which states that $\frac{d}{d p} [f (g (p))]$ is $f' (g (p)) g' (p)$ where $f (p) = p^{\frac{1}{2}}$ and $g (p) = 125 - p^{2}$ .

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Step 4.1.1

To apply the Chain Rule, set $u$ as $125 - p^{2}$ .

$\frac{d}{d u} [u^{\frac{1}{2}}] \frac{d}{d p} [125 - p^{2}]$

Step 4.1.2

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

$\frac{1}{2} u^{\frac{1}{2} - 1} \frac{d}{d p} [125 - p^{2}]$

Step 4.1.3

Replace all occurrences of $u$ with $125 - p^{2}$ .

$\frac{1}{2} {(125 - p^{2})}^{\frac{1}{2} - 1} \frac{d}{d p} [125 - p^{2}]$

Step 4.2

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

$\frac{1}{2} {(125 - p^{2})}^{\frac{1}{2} - 1 \cdot \frac{2}{2}} \frac{d}{d p} [125 - p^{2}]$

Step 4.3

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

$\frac{1}{2} {(125 - p^{2})}^{\frac{1}{2} + \frac{- 1 \cdot 2}{2}} \frac{d}{d p} [125 - p^{2}]$

Step 4.4

Combine the numerators over the common denominator.

$\frac{1}{2} {(125 - p^{2})}^{\frac{1 - 1 \cdot 2}{2}} \frac{d}{d p} [125 - p^{2}]$

Step 4.5

Simplify the numerator.

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Step 4.5.1

Multiply $- 1$ by $2$ .

$\frac{1}{2} {(125 - p^{2})}^{\frac{1 - 2}{2}} \frac{d}{d p} [125 - p^{2}]$

Step 4.5.2

Subtract $2$ from $1$ .

$\frac{1}{2} {(125 - p^{2})}^{\frac{- 1}{2}} \frac{d}{d p} [125 - p^{2}]$

Step 4.6

Combine fractions.

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Step 4.6.1

Move the negative in front of the fraction.

$\frac{1}{2} {(125 - p^{2})}^{- \frac{1}{2}} \frac{d}{d p} [125 - p^{2}]$

Step 4.6.2

Combine $\frac{1}{2}$ and ${(125 - p^{2})}^{- \frac{1}{2}}$ .

$\frac{{(125 - p^{2})}^{- \frac{1}{2}}}{2} \frac{d}{d p} [125 - p^{2}]$

Step 4.6.3

Move ${(125 - p^{2})}^{- \frac{1}{2}}$ to the denominator using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$\frac{1}{2 {(125 - p^{2})}^{\frac{1}{2}}} \frac{d}{d p} [125 - p^{2}]$

Step 4.7

By the Sum Rule, the derivative of $125 - p^{2}$ with respect to $p$ is $\frac{d}{d p} [125] + \frac{d}{d p} [- p^{2}]$ .

$\frac{1}{2 {(125 - p^{2})}^{\frac{1}{2}}} (\frac{d}{d p} [125] + \frac{d}{d p} [- p^{2}])$

Step 4.8

Since $125$ is constant with respect to $p$ , the derivative of $125$ with respect to $p$ is $0$ .

$\frac{1}{2 {(125 - p^{2})}^{\frac{1}{2}}} (0 + \frac{d}{d p} [- p^{2}])$

Step 4.9

Add $0$ and $\frac{d}{d p} [- p^{2}]$ .

$\frac{1}{2 {(125 - p^{2})}^{\frac{1}{2}}} \frac{d}{d p} [- p^{2}]$

Step 4.10

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

$\frac{1}{2 {(125 - p^{2})}^{\frac{1}{2}}} (- \frac{d}{d p} [p^{2}])$

Step 4.11

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

$\frac{1}{2 {(125 - p^{2})}^{\frac{1}{2}}} (- (2 p))$

Step 4.12

Simplify terms.

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Step 4.12.1

Multiply $2$ by $- 1$ .

$\frac{1}{2 {(125 - p^{2})}^{\frac{1}{2}}} (- 2 p)$

Step 4.12.2

Combine $- 2$ and $\frac{1}{2 {(125 - p^{2})}^{\frac{1}{2}}}$ .

$\frac{- 2}{2 {(125 - p^{2})}^{\frac{1}{2}}} p$

Step 4.12.3

Combine $\frac{- 2}{2 {(125 - p^{2})}^{\frac{1}{2}}}$ and $p$ .

$\frac{- 2 p}{2 {(125 - p^{2})}^{\frac{1}{2}}}$

Step 4.12.4

Factor $2$ out of $- 2 p$ .

$\frac{2 (- p)}{2 {(125 - p^{2})}^{\frac{1}{2}}}$

Step 4.13

Cancel the common factors.

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Step 4.13.1

Factor $2$ out of $2 {(125 - p^{2})}^{\frac{1}{2}}$ .

$\frac{2 (- p)}{2 ({(125 - p^{2})}^{\frac{1}{2}})}$

Step 4.13.2

Cancel the common factor.

$\frac{2 (- p)}{2 {(125 - p^{2})}^{\frac{1}{2}}}$

Step 4.13.3

Rewrite the expression.

$\frac{- p}{{(125 - p^{2})}^{\frac{1}{2}}}$

Step 4.14

Move the negative in front of the fraction.

$- \frac{p}{{(125 - p^{2})}^{\frac{1}{2}}}$

Step 5

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

$x' = - \frac{p}{{(125 - p^{2})}^{\frac{1}{2}}}$

Step 6

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

$\frac{d x}{d p} = - \frac{p}{{(125 - p^{2})}^{\frac{1}{2}}}$