Calculus Examples

$p = \frac{10000}{q^{2} + 25}$

Step 1

Differentiate both sides of the equation.

$\frac{d}{d q} (p) = \frac{d}{d q} (\frac{10000}{q^{2} + 25})$

Step 2

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

$p'$

Step 3

Differentiate the right side of the equation.

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Step 3.1

Differentiate using the Constant Multiple Rule.

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Step 3.1.1

Since $10000$ is constant with respect to $q$ , the derivative of $\frac{10000}{q^{2} + 25}$ with respect to $q$ is $10000 \frac{d}{d q} [\frac{1}{q^{2} + 25}]$ .

$10000 \frac{d}{d q} [\frac{1}{q^{2} + 25}]$

Step 3.1.2

Rewrite $\frac{1}{q^{2} + 25}$ as ${(q^{2} + 25)}^{- 1}$ .

$10000 \frac{d}{d q} [{(q^{2} + 25)}^{- 1}]$

Step 3.2

Differentiate using the chain rule, which states that $\frac{d}{d q} [f (g (q))]$ is $f' (g (q)) g' (q)$ where $f (q) = q^{- 1}$ and $g (q) = q^{2} + 25$ .

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Step 3.2.1

To apply the Chain Rule, set $u$ as $q^{2} + 25$ .

$10000 (\frac{d}{d u} [u^{- 1}] \frac{d}{d q} [q^{2} + 25])$

Step 3.2.2

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

$10000 (- u^{- 2} \frac{d}{d q} [q^{2} + 25])$

Step 3.2.3

Replace all occurrences of $u$ with $q^{2} + 25$ .

$10000 (- {(q^{2} + 25)}^{- 2} \frac{d}{d q} [q^{2} + 25])$

Step 3.3

Differentiate.

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Step 3.3.1

Multiply $- 1$ by $10000$ .

$- 10000 ({(q^{2} + 25)}^{- 2} \frac{d}{d q} [q^{2} + 25])$

Step 3.3.2

By the Sum Rule, the derivative of $q^{2} + 25$ with respect to $q$ is $\frac{d}{d q} [q^{2}] + \frac{d}{d q} [25]$ .

$- 10000 {(q^{2} + 25)}^{- 2} (\frac{d}{d q} [q^{2}] + \frac{d}{d q} [25])$

Step 3.3.3

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

$- 10000 {(q^{2} + 25)}^{- 2} (2 q + \frac{d}{d q} [25])$

Step 3.3.4

Since $25$ is constant with respect to $q$ , the derivative of $25$ with respect to $q$ is $0$ .

$- 10000 {(q^{2} + 25)}^{- 2} (2 q + 0)$

Step 3.3.5

Simplify the expression.

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Step 3.3.5.1

Add $2 q$ and $0$ .

$- 10000 {(q^{2} + 25)}^{- 2} (2 q)$

Step 3.3.5.2

Multiply $2$ by $- 10000$ .

$- 20000 {(q^{2} + 25)}^{- 2} q$

Step 3.4

Simplify.

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Step 3.4.1

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

$- 20000 \frac{1}{{(q^{2} + 25)}^{2}} q$

Step 3.4.2

Combine terms.

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Step 3.4.2.1

Combine $- 20000$ and $\frac{1}{{(q^{2} + 25)}^{2}}$ .

$\frac{- 20000}{{(q^{2} + 25)}^{2}} q$

Step 3.4.2.2

Move the negative in front of the fraction.

$- \frac{20000}{{(q^{2} + 25)}^{2}} q$

Step 3.4.2.3

Combine $q$ and $\frac{20000}{{(q^{2} + 25)}^{2}}$ .

$- \frac{q \cdot 20000}{{(q^{2} + 25)}^{2}}$

Step 3.4.2.4

Move $20000$ to the left of $q$ .

$- \frac{20000 q}{{(q^{2} + 25)}^{2}}$

Step 4

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

$p' = - \frac{20000 q}{{(q^{2} + 25)}^{2}}$

Step 5

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

$\frac{d p}{d q} = - \frac{20000 q}{{(q^{2} + 25)}^{2}}$