Calculus Examples

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

Step 1

Differentiate.

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Step 1.1

By the Sum Rule, the derivative of $1 + \frac{1}{x^{2}}$ with respect to $x$ is $\frac{d}{d x} [1] + \frac{d}{d x} [\frac{1}{x^{2}}]$ .

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

Step 1.2

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

$0 + \frac{d}{d x} [\frac{1}{x^{2}}]$

Step 2

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

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Step 2.1

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

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

Step 2.2

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

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Step 2.2.1

To apply the Chain Rule, set $u$ as $x^{2}$ .

$0 + \frac{d}{d u} [u^{- 1}] \frac{d}{d x} [x^{2}]$

Step 2.2.2

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

$0 - u^{- 2} \frac{d}{d x} [x^{2}]$

Step 2.2.3

Replace all occurrences of $u$ with $x^{2}$ .

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

Step 2.3

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

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

Step 2.4

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

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Step 2.4.1

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

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

Step 2.4.2

Multiply $2$ by $- 2$ .

$0 - x^{- 4} (2 x)$

Step 2.5

Multiply $2$ by $- 1$ .

$0 - 2 x^{- 4} x$

Step 2.6

Raise $x$ to the power of $1$ .

$0 - 2 (x^{1} x^{- 4})$

Step 2.7

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$0 - 2 x^{1 - 4}$

Step 2.8

Subtract $4$ from $1$ .

$0 - 2 x^{- 3}$

Step 3

Simplify.

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Step 3.1

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

$0 - 2 \frac{1}{x^{3}}$

Step 3.2

Combine terms.

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Step 3.2.1

Combine $- 2$ and $\frac{1}{x^{3}}$ .

$0 + \frac{- 2}{x^{3}}$

Step 3.2.2

Move the negative in front of the fraction.

$0 - \frac{2}{x^{3}}$

Step 3.2.3

Subtract $\frac{2}{x^{3}}$ from $0$ .

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