Calculus Examples

$\frac{π}{{(9 x)}^{2}}$

Step 1

Simplify with factoring out.

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Step 1.1

Factor $9$ out of $9 x$ .

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

Step 1.2

Simplify the expression.

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Step 1.2.1

Apply the product rule to $9 (x)$ .

$\frac{d}{d x} [\frac{π}{9^{2} x^{2}}]$

Step 1.2.2

Raise $9$ to the power of $2$ .

$\frac{d}{d x} [\frac{π}{81 x^{2}}]$

Step 2

Since $\frac{π}{81}$ is constant with respect to $x$ , the derivative of $\frac{π}{81 x^{2}}$ with respect to $x$ is $\frac{π}{81} \frac{d}{d x} [\frac{1}{x^{2}}]$ .

$\frac{π}{81} \frac{d}{d x} [\frac{1}{x^{2}}]$

Step 3

Apply basic rules of exponents.

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Step 3.1

Rewrite $\frac{1}{x^{2}}$ as ${(x^{2})}^{- 1}$ .

$\frac{π}{81} \frac{d}{d x} [{(x^{2})}^{- 1}]$

Step 3.2

Multiply the exponents in ${(x^{2})}^{- 1}$ .

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Step 3.2.1

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

$\frac{π}{81} \frac{d}{d x} [x^{2 \cdot - 1}]$

Step 3.2.2

Multiply $2$ by $- 1$ .

$\frac{π}{81} \frac{d}{d x} [x^{- 2}]$

Step 4

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

$\frac{π}{81} (- 2 x^{- 3})$

Step 5

Combine fractions.

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Step 5.1

Combine $- 2$ and $\frac{π}{81}$ .

$\frac{- 2 π}{81} x^{- 3}$

Step 5.2

Combine $\frac{- 2 π}{81}$ and $x^{- 3}$ .

$\frac{- 2 π x^{- 3}}{81}$

Step 5.3

Simplify the expression.

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Step 5.3.1

Move $x^{- 3}$ to the denominator using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$\frac{- 2 π}{81 x^{3}}$

Step 5.3.2

Move the negative in front of the fraction.

$- \frac{2 π}{81 x^{3}}$