Calculus Examples

$y = 9 x^{\frac{8}{3}}$

Step 1

Since $9$ is constant with respect to $x$ , the derivative of $9 x^{\frac{8}{3}}$ with respect to $x$ is $9 \frac{d}{d x} [x^{\frac{8}{3}}]$ .

$9 \frac{d}{d x} [x^{\frac{8}{3}}]$

Step 2

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = \frac{8}{3}$ .

$9 (\frac{8}{3} x^{\frac{8}{3} - 1})$

Step 3

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

$9 (\frac{8}{3} x^{\frac{8}{3} - 1 \cdot \frac{3}{3}})$

Step 4

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

$9 (\frac{8}{3} x^{\frac{8}{3} + \frac{- 1 \cdot 3}{3}})$

Step 5

Combine the numerators over the common denominator.

$9 (\frac{8}{3} x^{\frac{8 - 1 \cdot 3}{3}})$

Step 6

Simplify the numerator.

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Step 6.1

Multiply $- 1$ by $3$ .

$9 (\frac{8}{3} x^{\frac{8 - 3}{3}})$

Step 6.2

Subtract $3$ from $8$ .

$9 (\frac{8}{3} x^{\frac{5}{3}})$

Step 7

Combine $\frac{8}{3}$ and $x^{\frac{5}{3}}$ .

$9 \frac{8 x^{\frac{5}{3}}}{3}$

Step 8

Combine $9$ and $\frac{8 x^{\frac{5}{3}}}{3}$ .

$\frac{9 (8 x^{\frac{5}{3}})}{3}$

Step 9

Multiply $8$ by $9$ .

$\frac{72 x^{\frac{5}{3}}}{3}$

Step 10

Factor $3$ out of $72 x^{\frac{5}{3}}$ .

$\frac{3 (24 x^{\frac{5}{3}})}{3}$

Step 11

Cancel the common factors.

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Step 11.1

Factor $3$ out of $3$ .

$\frac{3 (24 x^{\frac{5}{3}})}{3 (1)}$

Step 11.2

Cancel the common factor.

$\frac{3 (24 x^{\frac{5}{3}})}{3 \cdot 1}$

Step 11.3

Rewrite the expression.

$\frac{24 x^{\frac{5}{3}}}{1}$

Step 11.4

Divide $24 x^{\frac{5}{3}}$ by $1$ .

$24 x^{\frac{5}{3}}$