Calculus Examples

$120 k^{\frac{1}{2}} l^{\frac{1}{3}}$

Step 1

Since $120 l^{\frac{1}{3}}$ is constant with respect to $k$ , the derivative of $120 k^{\frac{1}{2}} l^{\frac{1}{3}}$ with respect to $k$ is $120 l^{\frac{1}{3}} \frac{d}{d k} [k^{\frac{1}{2}}]$ .

$120 l^{\frac{1}{3}} \frac{d}{d k} [k^{\frac{1}{2}}]$

Step 2

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

$120 l^{\frac{1}{3}} (\frac{1}{2} k^{\frac{1}{2} - 1})$

Step 3

To write $- 1$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$120 l^{\frac{1}{3}} (\frac{1}{2} k^{\frac{1}{2} - 1 \cdot \frac{2}{2}})$

Step 4

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

$120 l^{\frac{1}{3}} (\frac{1}{2} k^{\frac{1}{2} + \frac{- 1 \cdot 2}{2}})$

Step 5

Combine the numerators over the common denominator.

$120 l^{\frac{1}{3}} (\frac{1}{2} k^{\frac{1 - 1 \cdot 2}{2}})$

Step 6

Simplify the numerator.

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Step 6.1

Multiply $- 1$ by $2$ .

$120 l^{\frac{1}{3}} (\frac{1}{2} k^{\frac{1 - 2}{2}})$

Step 6.2

Subtract $2$ from $1$ .

$120 l^{\frac{1}{3}} (\frac{1}{2} k^{\frac{- 1}{2}})$

Step 7

Move the negative in front of the fraction.

$120 l^{\frac{1}{3}} (\frac{1}{2} k^{- \frac{1}{2}})$

Step 8

Combine $\frac{1}{2}$ and $k^{- \frac{1}{2}}$ .

$120 l^{\frac{1}{3}} \frac{k^{- \frac{1}{2}}}{2}$

Step 9

Combine $120$ and $\frac{k^{- \frac{1}{2}}}{2}$ .

$l^{\frac{1}{3}} \frac{120 k^{- \frac{1}{2}}}{2}$

Step 10

Combine $l^{\frac{1}{3}}$ and $\frac{120 k^{- \frac{1}{2}}}{2}$ .

$\frac{l^{\frac{1}{3}} (120 k^{- \frac{1}{2}})}{2}$

Step 11

Simplify the expression.

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Step 11.1

Move $120$ to the left of $l^{\frac{1}{3}}$ .

$\frac{120 \cdot l^{\frac{1}{3}} k^{- \frac{1}{2}}}{2}$

Step 11.2

Move $k^{- \frac{1}{2}}$ to the denominator using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$\frac{120 l^{\frac{1}{3}}}{2 k^{\frac{1}{2}}}$

Step 12

Factor $2$ out of $120 l^{\frac{1}{3}}$ .

$\frac{2 (60 l^{\frac{1}{3}})}{2 k^{\frac{1}{2}}}$

Step 13

Cancel the common factors.

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Step 13.1

Factor $2$ out of $2 k^{\frac{1}{2}}$ .

$\frac{2 (60 l^{\frac{1}{3}})}{2 (k^{\frac{1}{2}})}$

Step 13.2

Cancel the common factor.

$\frac{2 (60 l^{\frac{1}{3}})}{2 k^{\frac{1}{2}}}$

Step 13.3

Rewrite the expression.

$\frac{60 l^{\frac{1}{3}}}{k^{\frac{1}{2}}}$