Calculus Examples

$20 L^{0.5} K^{0.5}$

Step 1

Since $20 L^{0.5}$ is constant with respect to $K$ , the derivative of $20 L^{0.5} K^{0.5}$ with respect to $K$ is $20 L^{0.5} \frac{d}{d K} [K^{0.5}]$ .

$20 L^{0.5} \frac{d}{d K} [K^{0.5}]$

Step 2

Differentiate using the Power Rule which states that $\frac{d}{d K} [K^{n}]$ is $n K^{n - 1}$ where $n = 0.5$ .

$20 L^{0.5} (0.5 K^{- 0.5})$

Step 3

Multiply $0.5$ by $20$ .

$10 L^{0.5} K^{- 0.5}$

Step 4

Simplify.

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Step 4.1

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

$10 L^{0.5} \frac{1}{K^{0.5}}$

Step 4.2

Combine terms.

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Step 4.2.1

Combine $10$ and $\frac{1}{K^{0.5}}$ .

$L^{0.5} \frac{10}{K^{0.5}}$

Step 4.2.2

Combine $L^{0.5}$ and $\frac{10}{K^{0.5}}$ .

$\frac{L^{0.5} \cdot 10}{K^{0.5}}$

Step 4.2.3

Move $10$ to the left of $L^{0.5}$ .

$\frac{10 L^{0.5}}{K^{0.5}}$