Calculus Examples

$r = \frac{4 e^{w}}{5 w}$

Step 1

Differentiate both sides of the equation.

$\frac{d}{d w} (r) = \frac{d}{d w} (\frac{4 e^{w}}{5 w})$

Step 2

The derivative of $r$ with respect to $w$ is $r'$ .

$r'$

Step 3

Differentiate the right side of the equation.

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Step 3.1

Since $\frac{4}{5}$ is constant with respect to $w$ , the derivative of $\frac{4 e^{w}}{5 w}$ with respect to $w$ is $\frac{4}{5} \frac{d}{d w} [\frac{e^{w}}{w}]$ .

$\frac{4}{5} \frac{d}{d w} [\frac{e^{w}}{w}]$

Step 3.2

Differentiate using the Quotient Rule which states that $\frac{d}{d w} [\frac{f (w)}{g (w)}]$ is $\frac{g (w) \frac{d}{d w} [f (w)] - f (w) \frac{d}{d w} [g (w)]}{{g (w)}^{2}}$ where $f (w) = e^{w}$ and $g (w) = w$ .

$\frac{4}{5} \cdot \frac{w \frac{d}{d w} [e^{w}] - e^{w} \frac{d}{d w} [w]}{w^{2}}$

Step 3.3

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

$\frac{4}{5} \cdot \frac{w e^{w} - e^{w} \frac{d}{d w} [w]}{w^{2}}$

Step 3.4

Differentiate using the Power Rule.

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Step 3.4.1

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

$\frac{4}{5} \cdot \frac{w e^{w} - e^{w} \cdot 1}{w^{2}}$

Step 3.4.2

Combine fractions.

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Step 3.4.2.1

Multiply $- 1$ by $1$ .

$\frac{4}{5} \cdot \frac{w e^{w} - e^{w}}{w^{2}}$

Step 3.4.2.2

Multiply $\frac{4}{5}$ by $\frac{w e^{w} - e^{w}}{w^{2}}$ .

$\frac{4 (w e^{w} - e^{w})}{5 w^{2}}$

Step 3.5

Simplify.

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Step 3.5.1

Apply the distributive property.

$\frac{4 (w e^{w}) + 4 (- e^{w})}{5 w^{2}}$

Step 3.5.2

Multiply $- 1$ by $4$ .

$\frac{4 w e^{w} - 4 e^{w}}{5 w^{2}}$

Step 3.5.3

Reorder terms.

$\frac{4 e^{w} w - 4 e^{w}}{5 w^{2}}$

Step 3.5.4

Factor $4 e^{w}$ out of $4 e^{w} w - 4 e^{w}$ .

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Step 3.5.4.1

Factor $4 e^{w}$ out of $4 e^{w} w$ .

$\frac{4 e^{w} (w) - 4 e^{w}}{5 w^{2}}$

Step 3.5.4.2

Factor $4 e^{w}$ out of $- 4 e^{w}$ .

$\frac{4 e^{w} (w) + 4 e^{w} (- 1)}{5 w^{2}}$

Step 3.5.4.3

Factor $4 e^{w}$ out of $4 e^{w} (w) + 4 e^{w} (- 1)$ .

$\frac{4 e^{w} (w - 1)}{5 w^{2}}$

Step 4

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

$r' = \frac{4 e^{w} (w - 1)}{5 w^{2}}$

Step 5

Replace $r'$ with $\frac{d r}{d w}$ .

$\frac{d r}{d w} = \frac{4 e^{w} (w - 1)}{5 w^{2}}$