Calculus Examples

$f (x) = \frac{x - 3}{x}$

Step 1

Find the first derivative.

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Step 1.1

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 $f (x) = x - 3$ and $g (x) = x$ .

$\frac{x \frac{d}{d x} [x - 3] - (x - 3) \frac{d}{d x} [x]}{x^{2}}$

Step 1.2

Differentiate.

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Step 1.2.1

By the Sum Rule, the derivative of $x - 3$ with respect to $x$ is $\frac{d}{d x} [x] + \frac{d}{d x} [- 3]$ .

$\frac{x (\frac{d}{d x} [x] + \frac{d}{d x} [- 3]) - (x - 3) \frac{d}{d x} [x]}{x^{2}}$

Step 1.2.2

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

$\frac{x (1 + \frac{d}{d x} [- 3]) - (x - 3) \frac{d}{d x} [x]}{x^{2}}$

Step 1.2.3

Since $- 3$ is constant with respect to $x$ , the derivative of $- 3$ with respect to $x$ is $0$ .

$\frac{x (1 + 0) - (x - 3) \frac{d}{d x} [x]}{x^{2}}$

Step 1.2.4

Simplify the expression.

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Step 1.2.4.1

Add $1$ and $0$ .

$\frac{x \cdot 1 - (x - 3) \frac{d}{d x} [x]}{x^{2}}$

Step 1.2.4.2

Multiply $x$ by $1$ .

$\frac{x - (x - 3) \frac{d}{d x} [x]}{x^{2}}$

Step 1.2.5

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

$\frac{x - (x - 3) \cdot 1}{x^{2}}$

Step 1.2.6

Multiply $- 1$ by $1$ .

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

Step 1.3

Simplify.

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Step 1.3.1

Apply the distributive property.

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

Step 1.3.2

Simplify the numerator.

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Step 1.3.2.1

Subtract $x$ from $x$ .

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

Step 1.3.2.2

Subtract $- 3$ from $0$ .

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

Step 1.3.2.3

Multiply $- 1$ by $- 3$ .

$f' (x) = \frac{3}{x^{2}}$

Step 2

Find the second derivative.

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Step 2.1

Since $3$ is constant with respect to $x$ , the derivative of $\frac{3}{x^{2}}$ with respect to $x$ is $3 \frac{d}{d x} [\frac{1}{x^{2}}]$ .

$3 \frac{d}{d x} [\frac{1}{x^{2}}]$

Step 2.2

Apply basic rules of exponents.

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Step 2.2.1

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

$3 \frac{d}{d x} [{(x^{2})}^{- 1}]$

Step 2.2.2

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

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Step 2.2.2.1

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

$3 \frac{d}{d x} [x^{2 \cdot - 1}]$

Step 2.2.2.2

Multiply $2$ by $- 1$ .

$3 \frac{d}{d x} [x^{- 2}]$

Step 2.3

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

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

Step 2.4

Multiply $- 2$ by $3$ .

$- 6 x^{- 3}$

Step 2.5

Simplify.

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Step 2.5.1

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

$- 6 \frac{1}{x^{3}}$

Step 2.5.2

Combine terms.

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Step 2.5.2.1

Combine $- 6$ and $\frac{1}{x^{3}}$ .

$\frac{- 6}{x^{3}}$

Step 2.5.2.2

Move the negative in front of the fraction.

$f'' (x) = - \frac{6}{x^{3}}$

Step 3

The second derivative of $f (x)$ with respect to $x$ is $- \frac{6}{x^{3}}$ .

$- \frac{6}{x^{3}}$