Calculus Examples

$f (x) = \frac{5}{x^{2}}$

Step 1

Since $5$ is constant with respect to $x$ , the derivative of $\frac{5}{x^{2}}$ with respect to $x$ is $5 \frac{d}{d x} [\frac{1}{x^{2}}]$ .

$5 \frac{d}{d x} [\frac{1}{x^{2}}]$

Step 2

Apply basic rules of exponents.

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Step 2.1

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

$5 \frac{d}{d x} [{(x^{2})}^{- 1}]$

Step 2.2

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

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Step 2.2.1

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

$5 \frac{d}{d x} [x^{2 \cdot - 1}]$

Step 2.2.2

Multiply $2$ by $- 1$ .

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

Step 3

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

$5 (- 2 x^{- 3})$

Step 4

Multiply $- 2$ by $5$ .

$- 10 x^{- 3}$

Step 5

Simplify.

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Step 5.1

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

$- 10 \frac{1}{x^{3}}$

Step 5.2

Combine terms.

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Step 5.2.1

Combine $- 10$ and $\frac{1}{x^{3}}$ .

$\frac{- 10}{x^{3}}$

Step 5.2.2

Move the negative in front of the fraction.

$- \frac{10}{x^{3}}$