Calculus Examples

$y = \frac{- 9}{{(3 x^{2})}^{3}}$

Step 1

Simplify terms.

Tap for more steps...

Step 1.1

Factor $3$ out of $3 x^{2}$ .

$\frac{d}{d x} [\frac{- 9}{{(3 (x^{2}))}^{3}}]$

Step 1.2

Simplify the expression.

Tap for more steps...

Step 1.2.1

Apply the product rule to $3 (x^{2})$ .

$\frac{d}{d x} [\frac{- 9}{3^{3} {(x^{2})}^{3}}]$

Step 1.2.2

Raise $3$ to the power of $3$ .

$\frac{d}{d x} [\frac{- 9}{27 {(x^{2})}^{3}}]$

Step 1.2.3

Multiply the exponents in ${(x^{2})}^{3}$ .

Tap for more steps...

Step 1.2.3.1

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$\frac{d}{d x} [\frac{- 9}{27 x^{2 \cdot 3}}]$

Step 1.2.3.2

Multiply $2$ by $3$ .

$\frac{d}{d x} [\frac{- 9}{27 x^{6}}]$

Step 1.3

Cancel the common factor of $- 9$ and $27$ .

Tap for more steps...

Step 1.3.1

Factor $9$ out of $- 9$ .

$\frac{d}{d x} [\frac{9 (- 1)}{27 x^{6}}]$

Step 1.3.2

Cancel the common factors.

Tap for more steps...

Step 1.3.2.1

Factor $9$ out of $27 x^{6}$ .

$\frac{d}{d x} [\frac{9 (- 1)}{9 (3 x^{6})}]$

Step 1.3.2.2

Cancel the common factor.

$\frac{d}{d x} [\frac{9 \cdot - 1}{9 (3 x^{6})}]$

Step 1.3.2.3

Rewrite the expression.

$\frac{d}{d x} [\frac{- 1}{3 x^{6}}]$

Step 1.4

Move the negative in front of the fraction.

$\frac{d}{d x} [- \frac{1}{3 x^{6}}]$

Step 2

Since $- \frac{1}{3}$ is constant with respect to $x$ , the derivative of $- \frac{1}{3 x^{6}}$ with respect to $x$ is $- \frac{1}{3} \frac{d}{d x} [\frac{1}{x^{6}}]$ .

$- \frac{1}{3} \frac{d}{d x} [\frac{1}{x^{6}}]$

Step 3

Apply basic rules of exponents.

Tap for more steps...

Step 3.1

Rewrite $\frac{1}{x^{6}}$ as ${(x^{6})}^{- 1}$ .

$- \frac{1}{3} \frac{d}{d x} [{(x^{6})}^{- 1}]$

Step 3.2

Multiply the exponents in ${(x^{6})}^{- 1}$ .

Tap for more steps...

Step 3.2.1

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$- \frac{1}{3} \frac{d}{d x} [x^{6 \cdot - 1}]$

Step 3.2.2

Multiply $6$ by $- 1$ .

$- \frac{1}{3} \frac{d}{d x} [x^{- 6}]$

Step 4

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = - 6$ .

$- \frac{1}{3} (- 6 x^{- 7})$

Step 5

Simplify terms.

Tap for more steps...

Step 5.1

Multiply $- 6$ by $- 1$ .

$6 (\frac{1}{3}) x^{- 7}$

Step 5.2

Combine $6$ and $\frac{1}{3}$ .

$\frac{6}{3} x^{- 7}$

Step 5.3

Combine $\frac{6}{3}$ and $x^{- 7}$ .

$\frac{6 x^{- 7}}{3}$

Step 5.4

Move $x^{- 7}$ to the denominator using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$\frac{6}{3 x^{7}}$

Step 5.5

Cancel the common factor of $6$ and $3$ .

Tap for more steps...

Step 5.5.1

Factor $3$ out of $6$ .

$\frac{3 \cdot 2}{3 x^{7}}$

Step 5.5.2

Cancel the common factors.

Tap for more steps...

Step 5.5.2.1

Factor $3$ out of $3 x^{7}$ .

$\frac{3 \cdot 2}{3 (x^{7})}$

Step 5.5.2.2

Cancel the common factor.

$\frac{3 \cdot 2}{3 x^{7}}$

Step 5.5.2.3

Rewrite the expression.

$\frac{2}{x^{7}}$