Calculus Examples

$q = l^{\frac{1}{4}}$

Step 1

Differentiate both sides of the equation.

$\frac{d}{d l} (q) = \frac{d}{d l} (l^{\frac{1}{4}})$

Step 2

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

$q'$

Step 3

Differentiate the right side of the equation.

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Step 3.1

Differentiate using the Power Rule which states that $\frac{d}{d l} [l^{n}]$ is $n l^{n - 1}$ where $n = \frac{1}{4}$ .

$\frac{1}{4} l^{\frac{1}{4} - 1}$

Step 3.2

To write $- 1$ as a fraction with a common denominator, multiply by $\frac{4}{4}$ .

$\frac{1}{4} l^{\frac{1}{4} - 1 \cdot \frac{4}{4}}$

Step 3.3

Combine $- 1$ and $\frac{4}{4}$ .

$\frac{1}{4} l^{\frac{1}{4} + \frac{- 1 \cdot 4}{4}}$

Step 3.4

Combine the numerators over the common denominator.

$\frac{1}{4} l^{\frac{1 - 1 \cdot 4}{4}}$

Step 3.5

Simplify the numerator.

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Step 3.5.1

Multiply $- 1$ by $4$ .

$\frac{1}{4} l^{\frac{1 - 4}{4}}$

Step 3.5.2

Subtract $4$ from $1$ .

$\frac{1}{4} l^{\frac{- 3}{4}}$

Step 3.6

Move the negative in front of the fraction.

$\frac{1}{4} l^{- \frac{3}{4}}$

Step 3.7

Simplify.

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Step 3.7.1

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

$\frac{1}{4} \cdot \frac{1}{l^{\frac{3}{4}}}$

Step 3.7.2

Multiply $\frac{1}{4}$ by $\frac{1}{l^{\frac{3}{4}}}$ .

$\frac{1}{4 l^{\frac{3}{4}}}$

Step 4

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

$q' = \frac{1}{4 l^{\frac{3}{4}}}$

Step 5

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

$\frac{d q}{d l} = \frac{1}{4 l^{\frac{3}{4}}}$