Calculus Examples

$\ln (\frac{1}{x^{2}})$

Step 1

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

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Step 1.1

To apply the Chain Rule, set $u$ as $\frac{1}{x^{2}}$ .

$\frac{d}{d u} [\ln (u)] \frac{d}{d x} [\frac{1}{x^{2}}]$

Step 1.2

The derivative of $\ln (u)$ with respect to $u$ is $\frac{1}{u}$ .

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

Step 1.3

Replace all occurrences of $u$ with $\frac{1}{x^{2}}$ .

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

Step 2

Multiply by the reciprocal of the fraction to divide by $\frac{1}{x^{2}}$ .

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

Step 3

Simplify the expression.

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Step 3.1

Multiply $x^{2}$ by $1$ .

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

Step 3.2

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

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

Step 3.3

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

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Step 3.3.1

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

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

Step 3.3.2

Multiply $2$ by $- 1$ .

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

Step 4

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

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

Step 5

Multiply $x^{2}$ by $x^{- 3}$ by adding the exponents.

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Step 5.1

Move $x^{- 3}$ .

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

Step 5.2

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

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

Step 5.3

Add $- 3$ and $2$ .

$x^{- 1} \cdot - 2$

Step 6

Move $- 2$ to the left of $x^{- 1}$ .

$- 2 \cdot x^{- 1}$

Step 7

Simplify.

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Step 7.1

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

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

Step 7.2

Combine terms.

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Step 7.2.1

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

$\frac{- 2}{x}$

Step 7.2.2

Move the negative in front of the fraction.

$- \frac{2}{x}$