Calculus Examples

$x^{\frac{5}{3} - 20} x^{\frac{2}{3}}$

Step 1

To write $- 20$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

$\frac{d}{d x} [x^{\frac{5}{3} - 20 \cdot \frac{3}{3}} x^{\frac{2}{3}}]$

Step 2

Combine $- 20$ and $\frac{3}{3}$ .

$\frac{d}{d x} [x^{\frac{5}{3} + \frac{- 20 \cdot 3}{3}} x^{\frac{2}{3}}]$

Step 3

Combine the numerators over the common denominator.

$\frac{d}{d x} [x^{\frac{5 - 20 \cdot 3}{3}} x^{\frac{2}{3}}]$

Step 4

Simplify the numerator.

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Step 4.1

Multiply $- 20$ by $3$ .

$\frac{d}{d x} [x^{\frac{5 - 60}{3}} x^{\frac{2}{3}}]$

Step 4.2

Subtract $60$ from $5$ .

$\frac{d}{d x} [x^{\frac{- 55}{3}} x^{\frac{2}{3}}]$

Step 5

Move the negative in front of the fraction.

$\frac{d}{d x} [x^{- \frac{55}{3}} x^{\frac{2}{3}}]$

Step 6

Multiply $x^{- \frac{55}{3}}$ by $x^{\frac{2}{3}}$ by adding the exponents.

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Step 6.1

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$\frac{d}{d x} [x^{- \frac{55}{3} + \frac{2}{3}}]$

Step 6.2

Combine the numerators over the common denominator.

$\frac{d}{d x} [x^{\frac{- 55 + 2}{3}}]$

Step 6.3

Add $- 55$ and $2$ .

$\frac{d}{d x} [x^{\frac{- 53}{3}}]$

Step 6.4

Move the negative in front of the fraction.

$\frac{d}{d x} [x^{- \frac{53}{3}}]$

Step 7

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = - \frac{53}{3}$ .

$- \frac{53}{3} x^{- \frac{53}{3} - 1}$

Step 8

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

$- \frac{53}{3} x^{- \frac{53}{3} - 1 \cdot \frac{3}{3}}$

Step 9

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

$- \frac{53}{3} x^{- \frac{53}{3} + \frac{- 1 \cdot 3}{3}}$

Step 10

Combine the numerators over the common denominator.

$- \frac{53}{3} x^{\frac{- 53 - 1 \cdot 3}{3}}$

Step 11

Simplify the numerator.

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Step 11.1

Multiply $- 1$ by $3$ .

$- \frac{53}{3} x^{\frac{- 53 - 3}{3}}$

Step 11.2

Subtract $3$ from $- 53$ .

$- \frac{53}{3} x^{\frac{- 56}{3}}$

Step 12

Move the negative in front of the fraction.

$- \frac{53}{3} x^{- \frac{56}{3}}$

Step 13

Simplify.

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Step 13.1

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

$- \frac{53}{3} \cdot \frac{1}{x^{\frac{56}{3}}}$

Step 13.2

Combine terms.

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Step 13.2.1

Multiply $\frac{1}{x^{\frac{56}{3}}}$ by $\frac{53}{3}$ .

$- \frac{53}{x^{\frac{56}{3}} \cdot 3}$

Step 13.2.2

Move $3$ to the left of $x^{\frac{56}{3}}$ .

$- \frac{53}{3 x^{\frac{56}{3}}}$