Calculus Examples

$p = 500 e^{- 0.4 q}$

Step 1

Differentiate both sides of the equation.

$\frac{d}{d p} (p) = \frac{d}{d p} (500 e^{- 0.4 q})$

Step 2

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

$1$

Step 3

Differentiate the right side of the equation.

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Step 3.1

Since $500$ is constant with respect to $p$ , the derivative of $500 e^{- 0.4 q}$ with respect to $p$ is $500 \frac{d}{d p} [e^{- 0.4 q}]$ .

$500 \frac{d}{d p} [e^{- 0.4 q}]$

Step 3.2

Differentiate using the chain rule, which states that $\frac{d}{d p} [f (g (p))]$ is $f' (g (p)) g' (p)$ where $f (p) = e^{p}$ and $g (p) = - 0.4 q$ .

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Step 3.2.1

To apply the Chain Rule, set $u$ as $- 0.4 q$ .

$500 (\frac{d}{d u} [e^{u}] \frac{d}{d p} [- 0.4 q])$

Step 3.2.2

Differentiate using the Exponential Rule which states that $\frac{d}{d u} [a^{u}]$ is $a^{u} \ln (a)$ where $a$ = $e$ .

$500 (e^{u} \frac{d}{d p} [- 0.4 q])$

Step 3.2.3

Replace all occurrences of $u$ with $- 0.4 q$ .

$500 (e^{- 0.4 q} \frac{d}{d p} [- 0.4 q])$

Step 3.3

Differentiate using the Constant Multiple Rule.

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Step 3.3.1

Since $- 0.4$ is constant with respect to $p$ , the derivative of $- 0.4 q$ with respect to $p$ is $- 0.4 \frac{d}{d p} [q]$ .

$500 e^{- 0.4 q} (- 0.4 \frac{d}{d p} [q])$

Step 3.3.2

Multiply $- 0.4$ by $500$ .

$- 200 e^{- 0.4 q} \frac{d}{d p} [q]$

Step 3.4

Rewrite $\frac{d}{d p} [q]$ as $q'$ .

$- 200 e^{- 0.4 q} q'$

Step 4

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

$1 = - 200 e^{- 0.4 q} q'$

Step 5

Solve for $q'$ .

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Step 5.1

Rewrite the equation as $- 200 e^{- 0.4 q} q' = 1$ .

$- 200 e^{- 0.4 q} q' = 1$

Step 5.2

Divide each term in $- 200 e^{- 0.4 q} q' = 1$ by $- 200 e^{- 0.4 q}$ and simplify.

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Step 5.2.1

Divide each term in $- 200 e^{- 0.4 q} q' = 1$ by $- 200 e^{- 0.4 q}$ .

$\frac{- 200 e^{- 0.4 q} q'}{- 200 e^{- 0.4 q}} = \frac{1}{- 200 e^{- 0.4 q}}$

Step 5.2.2

Simplify the left side.

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Step 5.2.2.1

Cancel the common factor of $- 200$ .

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Step 5.2.2.1.1

Cancel the common factor.

$\frac{- 200 e^{- 0.4 q} q'}{- 200 e^{- 0.4 q}} = \frac{1}{- 200 e^{- 0.4 q}}$

Step 5.2.2.1.2

Rewrite the expression.

$\frac{e^{- 0.4 q} q'}{e^{- 0.4 q}} = \frac{1}{- 200 e^{- 0.4 q}}$

Step 5.2.2.2

Cancel the common factor of $e^{- 0.4 q}$ .

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Step 5.2.2.2.1

Cancel the common factor.

$\frac{e^{- 0.4 q} q'}{e^{- 0.4 q}} = \frac{1}{- 200 e^{- 0.4 q}}$

Step 5.2.2.2.2

Divide $q'$ by $1$ .

$q' = \frac{1}{- 200 e^{- 0.4 q}}$

Step 5.2.3

Simplify the right side.

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Step 5.2.3.1

Move the negative in front of the fraction.

$q' = - \frac{1}{200 e^{- 0.4 q}}$

Step 5.2.3.2

Rewrite $1$ as $1 (1)$ .

$q' = - \frac{1 (1)}{200 e^{- 0.4 q}}$

Step 5.2.3.3

Factor $200$ out of $200 e^{- 0.4 q}$ .

$q' = - \frac{1 (1)}{200 (e^{- 0.4 q})}$

Step 5.2.3.4

Separate fractions.

$q' = - (\frac{1}{200} \cdot \frac{1}{e^{- 0.4 q}})$

Step 5.2.3.5

Divide $1$ by $200$ .

$q' = - (0.005 \frac{1}{e^{- 0.4 q}})$

Step 5.2.3.6

Combine $0.005$ and $\frac{1}{e^{- 0.4 q}}$ .

$q' = - \frac{0.005}{e^{- 0.4 q}}$

Step 6

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

$\frac{d q}{d p} = - \frac{0.005}{e^{- 0.4 q}}$