Calculus Examples

$\frac{1}{x^{2}}$

Step 1

Differentiate using the Quotient Rule which states that $\frac{d}{d x} [\frac{f (x)}{g (x)}]$ is $\frac{g (x) \frac{d}{d x} [f (x)] - f (x) \frac{d}{d x} [g (x)]}{{g (x)}^{2}}$ where $f (x) = 1$ and $g (x) = x^{2}$ .

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

Step 2

Differentiate.

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Step 2.1

Since $1$ is constant with respect to $x$ , the derivative of $1$ with respect to $x$ is $0$ .

$\frac{x^{2} \cdot 0 - 1 \cdot 1 \frac{d}{d x} [x^{2}]}{{(x^{2})}^{2}}$

Step 2.2

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

$\frac{x^{2} \cdot 0 - 1 \cdot 1 (2 x)}{{(x^{2})}^{2}}$

Step 3

Simplify.

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Step 3.1

Simplify the numerator.

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Step 3.1.1

Simplify each term.

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Step 3.1.1.1

Multiply $x^{2}$ by $0$ .

$\frac{0 - 1 \cdot 1 (2 x)}{{(x^{2})}^{2}}$

Step 3.1.1.2

Multiply $- 1$ by $1$ .

$\frac{0 - (2 x)}{{(x^{2})}^{2}}$

Step 3.1.1.3

Multiply $2$ by $- 1$ .

$\frac{0 - 2 x}{{(x^{2})}^{2}}$

Step 3.1.2

Subtract $2 x$ from $0$ .

$\frac{- 2 x}{{(x^{2})}^{2}}$

Step 3.2

Combine terms.

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Step 3.2.1

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

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Step 3.2.1.1

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

$\frac{- 2 x}{x^{2 \cdot 2}}$

Step 3.2.1.2

Multiply $2$ by $2$ .

$\frac{- 2 x}{x^{4}}$

Step 3.2.2

Cancel the common factor of $x$ and $x^{4}$ .

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Step 3.2.2.1

Factor $x$ out of $- 2 x$ .

$\frac{x \cdot - 2}{x^{4}}$

Step 3.2.2.2

Cancel the common factors.

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Step 3.2.2.2.1

Factor $x$ out of $x^{4}$ .

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

Step 3.2.2.2.2

Cancel the common factor.

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

Step 3.2.2.2.3

Rewrite the expression.

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

Step 3.2.3

Move the negative in front of the fraction.

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