Calculus Examples

$40 x^{\frac{4}{5}} y^{\frac{1}{5}}$

Step 1

Since $40 y^{\frac{1}{5}}$ is constant with respect to $x$ , the derivative of $40 x^{\frac{4}{5}} y^{\frac{1}{5}}$ with respect to $x$ is $40 y^{\frac{1}{5}} \frac{d}{d x} [x^{\frac{4}{5}}]$ .

$40 y^{\frac{1}{5}} \frac{d}{d x} [x^{\frac{4}{5}}]$

Step 2

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = \frac{4}{5}$ .

$40 y^{\frac{1}{5}} (\frac{4}{5} x^{\frac{4}{5} - 1})$

Step 3

To write $- 1$ as a fraction with a common denominator, multiply by $\frac{5}{5}$ .

$40 y^{\frac{1}{5}} (\frac{4}{5} x^{\frac{4}{5} - 1 \cdot \frac{5}{5}})$

Step 4

Combine $- 1$ and $\frac{5}{5}$ .

$40 y^{\frac{1}{5}} (\frac{4}{5} x^{\frac{4}{5} + \frac{- 1 \cdot 5}{5}})$

Step 5

Combine the numerators over the common denominator.

$40 y^{\frac{1}{5}} (\frac{4}{5} x^{\frac{4 - 1 \cdot 5}{5}})$

Step 6

Simplify the numerator.

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Step 6.1

Multiply $- 1$ by $5$ .

$40 y^{\frac{1}{5}} (\frac{4}{5} x^{\frac{4 - 5}{5}})$

Step 6.2

Subtract $5$ from $4$ .

$40 y^{\frac{1}{5}} (\frac{4}{5} x^{\frac{- 1}{5}})$

Step 7

Move the negative in front of the fraction.

$40 y^{\frac{1}{5}} (\frac{4}{5} x^{- \frac{1}{5}})$

Step 8

Combine $\frac{4}{5}$ and $x^{- \frac{1}{5}}$ .

$40 y^{\frac{1}{5}} \frac{4 x^{- \frac{1}{5}}}{5}$

Step 9

Combine $40$ and $\frac{4 x^{- \frac{1}{5}}}{5}$ .

$y^{\frac{1}{5}} \frac{40 (4 x^{- \frac{1}{5}})}{5}$

Step 10

Multiply $4$ by $40$ .

$y^{\frac{1}{5}} \frac{160 x^{- \frac{1}{5}}}{5}$

Step 11

Combine $y^{\frac{1}{5}}$ and $\frac{160 x^{- \frac{1}{5}}}{5}$ .

$\frac{y^{\frac{1}{5}} (160 x^{- \frac{1}{5}})}{5}$

Step 12

Simplify the expression.

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Step 12.1

Move $160$ to the left of $y^{\frac{1}{5}}$ .

$\frac{160 \cdot y^{\frac{1}{5}} x^{- \frac{1}{5}}}{5}$

Step 12.2

Move $x^{- \frac{1}{5}}$ to the denominator using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$\frac{160 y^{\frac{1}{5}}}{5 x^{\frac{1}{5}}}$

Step 13

Factor $5$ out of $160 y^{\frac{1}{5}}$ .

$\frac{5 (32 y^{\frac{1}{5}})}{5 x^{\frac{1}{5}}}$

Step 14

Cancel the common factors.

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Step 14.1

Factor $5$ out of $5 x^{\frac{1}{5}}$ .

$\frac{5 (32 y^{\frac{1}{5}})}{5 (x^{\frac{1}{5}})}$

Step 14.2

Cancel the common factor.

$\frac{5 (32 y^{\frac{1}{5}})}{5 x^{\frac{1}{5}}}$

Step 14.3

Rewrite the expression.

$\frac{32 y^{\frac{1}{5}}}{x^{\frac{1}{5}}}$