Calculus Examples

$K = \frac{1}{L^{\frac{2}{3}}}$

Step 1

Differentiate both sides of the equation.

$\frac{d}{d L} (K) = \frac{d}{d L} (\frac{1}{L^{\frac{2}{3}}})$

Step 2

The derivative of $K$ with respect to $L$ is $K'$ .

$K'$

Step 3

Differentiate the right side of the equation.

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Step 3.1

Apply basic rules of exponents.

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Step 3.1.1

Rewrite $\frac{1}{L^{\frac{2}{3}}}$ as ${(L^{\frac{2}{3}})}^{- 1}$ .

$\frac{d}{d L} [{(L^{\frac{2}{3}})}^{- 1}]$

Step 3.1.2

Multiply the exponents in ${(L^{\frac{2}{3}})}^{- 1}$ .

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Step 3.1.2.1

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

$\frac{d}{d L} [L^{\frac{2}{3} \cdot - 1}]$

Step 3.1.2.2

Multiply $\frac{2}{3} \cdot - 1$ .

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Step 3.1.2.2.1

Combine $\frac{2}{3}$ and $- 1$ .

$\frac{d}{d L} [L^{\frac{2 \cdot - 1}{3}}]$

Step 3.1.2.2.2

Multiply $2$ by $- 1$ .

$\frac{d}{d L} [L^{\frac{- 2}{3}}]$

Step 3.1.2.3

Move the negative in front of the fraction.

$\frac{d}{d L} [L^{- \frac{2}{3}}]$

Step 3.2

Differentiate using the Power Rule which states that $\frac{d}{d L} [L^{n}]$ is $n L^{n - 1}$ where $n = - \frac{2}{3}$ .

$- \frac{2}{3} L^{- \frac{2}{3} - 1}$

Step 3.3

To write $- 1$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

$- \frac{2}{3} L^{- \frac{2}{3} - 1 \cdot \frac{3}{3}}$

Step 3.4

Combine $- 1$ and $\frac{3}{3}$ .

$- \frac{2}{3} L^{- \frac{2}{3} + \frac{- 1 \cdot 3}{3}}$

Step 3.5

Combine the numerators over the common denominator.

$- \frac{2}{3} L^{\frac{- 2 - 1 \cdot 3}{3}}$

Step 3.6

Simplify the numerator.

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Step 3.6.1

Multiply $- 1$ by $3$ .

$- \frac{2}{3} L^{\frac{- 2 - 3}{3}}$

Step 3.6.2

Subtract $3$ from $- 2$ .

$- \frac{2}{3} L^{\frac{- 5}{3}}$

Step 3.7

Move the negative in front of the fraction.

$- \frac{2}{3} L^{- \frac{5}{3}}$

Step 3.8

Simplify.

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Step 3.8.1

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

$- \frac{2}{3} \cdot \frac{1}{L^{\frac{5}{3}}}$

Step 3.8.2

Combine terms.

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Step 3.8.2.1

Multiply $\frac{1}{L^{\frac{5}{3}}}$ by $\frac{2}{3}$ .

$- \frac{2}{L^{\frac{5}{3}} \cdot 3}$

Step 3.8.2.2

Move $3$ to the left of $L^{\frac{5}{3}}$ .

$- \frac{2}{3 L^{\frac{5}{3}}}$

Step 4

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

$K' = - \frac{2}{3 L^{\frac{5}{3}}}$

Step 5

Replace $K'$ with $\frac{d K}{d L}$ .

$\frac{d K}{d L} = - \frac{2}{3 L^{\frac{5}{3}}}$