Calculus Examples

$c = 0.001 q^{2} - 0.2 q + 11 + \frac{15000}{q}$

Step 1

Differentiate both sides of the equation.

$\frac{d}{d q} (c) = \frac{d}{d q} (0.001 q^{2} - 0.2 q + 11 + \frac{15000}{q})$

Step 2

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

$c'$

Step 3

Differentiate the right side of the equation.

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Step 3.1

By the Sum Rule, the derivative of $0.001 q^{2} - 0.2 q + 11 + \frac{15000}{q}$ with respect to $q$ is $\frac{d}{d q} [0.001 q^{2}] + \frac{d}{d q} [- 0.2 q] + \frac{d}{d q} [11] + \frac{d}{d q} [\frac{15000}{q}]$ .

$\frac{d}{d q} [0.001 q^{2}] + \frac{d}{d q} [- 0.2 q] + \frac{d}{d q} [11] + \frac{d}{d q} [\frac{15000}{q}]$

Step 3.2

Evaluate $\frac{d}{d q} [0.001 q^{2}]$ .

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Step 3.2.1

Since $0.001$ is constant with respect to $q$ , the derivative of $0.001 q^{2}$ with respect to $q$ is $0.001 \frac{d}{d q} [q^{2}]$ .

$0.001 \frac{d}{d q} [q^{2}] + \frac{d}{d q} [- 0.2 q] + \frac{d}{d q} [11] + \frac{d}{d q} [\frac{15000}{q}]$

Step 3.2.2

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

$0.001 (2 q) + \frac{d}{d q} [- 0.2 q] + \frac{d}{d q} [11] + \frac{d}{d q} [\frac{15000}{q}]$

Step 3.2.3

Multiply $2$ by $0.001$ .

$0.002 q + \frac{d}{d q} [- 0.2 q] + \frac{d}{d q} [11] + \frac{d}{d q} [\frac{15000}{q}]$

Step 3.3

Evaluate $\frac{d}{d q} [- 0.2 q]$ .

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Step 3.3.1

Since $- 0.2$ is constant with respect to $q$ , the derivative of $- 0.2 q$ with respect to $q$ is $- 0.2 \frac{d}{d q} [q]$ .

$0.002 q - 0.2 \frac{d}{d q} [q] + \frac{d}{d q} [11] + \frac{d}{d q} [\frac{15000}{q}]$

Step 3.3.2

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

$0.002 q - 0.2 \cdot 1 + \frac{d}{d q} [11] + \frac{d}{d q} [\frac{15000}{q}]$

Step 3.3.3

Multiply $- 0.2$ by $1$ .

$0.002 q - 0.2 + \frac{d}{d q} [11] + \frac{d}{d q} [\frac{15000}{q}]$

Step 3.4

Since $11$ is constant with respect to $q$ , the derivative of $11$ with respect to $q$ is $0$ .

$0.002 q - 0.2 + 0 + \frac{d}{d q} [\frac{15000}{q}]$

Step 3.5

Evaluate $\frac{d}{d q} [\frac{15000}{q}]$ .

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Step 3.5.1

Since $15000$ is constant with respect to $q$ , the derivative of $\frac{15000}{q}$ with respect to $q$ is $15000 \frac{d}{d q} [\frac{1}{q}]$ .

$0.002 q - 0.2 + 0 + 15000 \frac{d}{d q} [\frac{1}{q}]$

Step 3.5.2

Rewrite $\frac{1}{q}$ as $q^{- 1}$ .

$0.002 q - 0.2 + 0 + 15000 \frac{d}{d q} [q^{- 1}]$

Step 3.5.3

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

$0.002 q - 0.2 + 0 + 15000 (- q^{- 2})$

Step 3.5.4

Multiply $- 1$ by $15000$ .

$0.002 q - 0.2 + 0 - 15000 q^{- 2}$

Step 3.6

Simplify.

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Step 3.6.1

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

$0.002 q - 0.2 + 0 - 15000 \frac{1}{q^{2}}$

Step 3.6.2

Combine terms.

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Step 3.6.2.1

Add $0.002 q - 0.2$ and $0$ .

$0.002 q - 0.2 - 15000 \frac{1}{q^{2}}$

Step 3.6.2.2

Combine $- 15000$ and $\frac{1}{q^{2}}$ .

$0.002 q - 0.2 + \frac{- 15000}{q^{2}}$

Step 3.6.2.3

Move the negative in front of the fraction.

$0.002 q - 0.2 - \frac{15000}{q^{2}}$

Step 3.6.3

Reorder terms.

$0.002 q - \frac{15000}{q^{2}} - 0.2$

Step 4

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

$c' = 0.002 q - \frac{15000}{q^{2}} - 0.2$

Step 5

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

$\frac{d c}{d q} = 0.002 q - \frac{15000}{q^{2}} - 0.2$