Calculus Examples

$\frac{w - 5}{w^{5}}$

Step 1

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 $f (w) = w - 5$ and $g (w) = w^{5}$ .

$\frac{w^{5} \frac{d}{d w} [w - 5] - (w - 5) \frac{d}{d w} [w^{5}]}{{(w^{5})}^{2}}$

Step 2

Differentiate.

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Step 2.1

Multiply the exponents in ${(w^{5})}^{2}$ .

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Step 2.1.1

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$\frac{w^{5} \frac{d}{d w} [w - 5] - (w - 5) \frac{d}{d w} [w^{5}]}{w^{5 \cdot 2}}$

Step 2.1.2

Multiply $5$ by $2$ .

$\frac{w^{5} \frac{d}{d w} [w - 5] - (w - 5) \frac{d}{d w} [w^{5}]}{w^{10}}$

Step 2.2

By the Sum Rule, the derivative of $w - 5$ with respect to $w$ is $\frac{d}{d w} [w] + \frac{d}{d w} [- 5]$ .

$\frac{w^{5} (\frac{d}{d w} [w] + \frac{d}{d w} [- 5]) - (w - 5) \frac{d}{d w} [w^{5}]}{w^{10}}$

Step 2.3

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

$\frac{w^{5} (1 + \frac{d}{d w} [- 5]) - (w - 5) \frac{d}{d w} [w^{5}]}{w^{10}}$

Step 2.4

Since $- 5$ is constant with respect to $w$ , the derivative of $- 5$ with respect to $w$ is $0$ .

$\frac{w^{5} (1 + 0) - (w - 5) \frac{d}{d w} [w^{5}]}{w^{10}}$

Step 2.5

Simplify the expression.

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Step 2.5.1

Add $1$ and $0$ .

$\frac{w^{5} \cdot 1 - (w - 5) \frac{d}{d w} [w^{5}]}{w^{10}}$

Step 2.5.2

Multiply $w^{5}$ by $1$ .

$\frac{w^{5} - (w - 5) \frac{d}{d w} [w^{5}]}{w^{10}}$

Step 2.6

Differentiate using the Power Rule which states that $\frac{d}{d w} [w^{n}]$ is $n w^{n - 1}$ where $n = 5$ .

$\frac{w^{5} - (w - 5) (5 w^{4})}{w^{10}}$

Step 2.7

Simplify with factoring out.

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Step 2.7.1

Multiply $5$ by $- 1$ .

$\frac{w^{5} - 5 (w - 5) w^{4}}{w^{10}}$

Step 2.7.2

Factor $w^{4}$ out of $w^{5} - 5 (w - 5) w^{4}$ .

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Step 2.7.2.1

Factor $w^{4}$ out of $w^{5}$ .

$\frac{w^{4} w - 5 (w - 5) w^{4}}{w^{10}}$

Step 2.7.2.2

Factor $w^{4}$ out of $- 5 (w - 5) w^{4}$ .

$\frac{w^{4} w + w^{4} (- 5 (w - 5))}{w^{10}}$

Step 2.7.2.3

Factor $w^{4}$ out of $w^{4} w + w^{4} (- 5 (w - 5))$ .

$\frac{w^{4} (w - 5 (w - 5))}{w^{10}}$

Step 3

Cancel the common factors.

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Step 3.1

Factor $w^{4}$ out of $w^{10}$ .

$\frac{w^{4} (w - 5 (w - 5))}{w^{4} w^{6}}$

Step 3.2

Cancel the common factor.

$\frac{w^{4} (w - 5 (w - 5))}{w^{4} w^{6}}$

Step 3.3

Rewrite the expression.

$\frac{w - 5 (w - 5)}{w^{6}}$

Step 4

Simplify.

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Step 4.1

Apply the distributive property.

$\frac{w - 5 w - 5 \cdot - 5}{w^{6}}$

Step 4.2

Simplify the numerator.

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Step 4.2.1

Multiply $- 5$ by $- 5$ .

$\frac{w - 5 w + 25}{w^{6}}$

Step 4.2.2

Subtract $5 w$ from $w$ .

$\frac{- 4 w + 25}{w^{6}}$

Step 4.3

Factor $- 1$ out of $- 4 w$ .

$\frac{- (4 w) + 25}{w^{6}}$

Step 4.4

Rewrite $25$ as $- 1 (- 25)$ .

$\frac{- (4 w) - 1 (- 25)}{w^{6}}$

Step 4.5

Factor $- 1$ out of $- (4 w) - 1 (- 25)$ .

$\frac{- (4 w - 25)}{w^{6}}$

Step 4.6

Rewrite $- (4 w - 25)$ as $- 1 (4 w - 25)$ .

$\frac{- 1 (4 w - 25)}{w^{6}}$

Step 4.7

Move the negative in front of the fraction.

$- \frac{4 w - 25}{w^{6}}$