Calculus Examples

$y = \frac{x - 6}{4 x}$

Step 1

Find the first derivative.

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Step 1.1

Since $\frac{1}{4}$ is constant with respect to $x$ , the derivative of $\frac{x - 6}{4 x}$ with respect to $x$ is $\frac{1}{4} \frac{d}{d x} [\frac{x - 6}{x}]$ .

$\frac{1}{4} \frac{d}{d x} [\frac{x - 6}{x}]$

Step 1.2

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 $f (x) = x - 6$ and $g (x) = x$ .

$\frac{1}{4} \cdot \frac{x \frac{d}{d x} [x - 6] - (x - 6) \frac{d}{d x} [x]}{x^{2}}$

Step 1.3

Differentiate.

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Step 1.3.1

By the Sum Rule, the derivative of $x - 6$ with respect to $x$ is $\frac{d}{d x} [x] + \frac{d}{d x} [- 6]$ .

$\frac{1}{4} \cdot \frac{x (\frac{d}{d x} [x] + \frac{d}{d x} [- 6]) - (x - 6) \frac{d}{d x} [x]}{x^{2}}$

Step 1.3.2

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

$\frac{1}{4} \cdot \frac{x (1 + \frac{d}{d x} [- 6]) - (x - 6) \frac{d}{d x} [x]}{x^{2}}$

Step 1.3.3

Since $- 6$ is constant with respect to $x$ , the derivative of $- 6$ with respect to $x$ is $0$ .

$\frac{1}{4} \cdot \frac{x (1 + 0) - (x - 6) \frac{d}{d x} [x]}{x^{2}}$

Step 1.3.4

Simplify the expression.

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Step 1.3.4.1

Add $1$ and $0$ .

$\frac{1}{4} \cdot \frac{x \cdot 1 - (x - 6) \frac{d}{d x} [x]}{x^{2}}$

Step 1.3.4.2

Multiply $x$ by $1$ .

$\frac{1}{4} \cdot \frac{x - (x - 6) \frac{d}{d x} [x]}{x^{2}}$

Step 1.3.5

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

$\frac{1}{4} \cdot \frac{x - (x - 6) \cdot 1}{x^{2}}$

Step 1.3.6

Combine fractions.

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Step 1.3.6.1

Multiply $- 1$ by $1$ .

$\frac{1}{4} \cdot \frac{x - (x - 6)}{x^{2}}$

Step 1.3.6.2

Multiply $\frac{1}{4}$ by $\frac{x - (x - 6)}{x^{2}}$ .

$\frac{x - (x - 6)}{4 x^{2}}$

Step 1.4

Simplify.

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Step 1.4.1

Apply the distributive property.

$\frac{x - x - - 6}{4 x^{2}}$

Step 1.4.2

Simplify the numerator.

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Step 1.4.2.1

Subtract $x$ from $x$ .

$\frac{0 - - 6}{4 x^{2}}$

Step 1.4.2.2

Subtract $- 6$ from $0$ .

$\frac{- - 6}{4 x^{2}}$

Step 1.4.2.3

Multiply $- 1$ by $- 6$ .

$\frac{6}{4 x^{2}}$

Step 1.4.3

Cancel the common factor of $6$ and $4$ .

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Step 1.4.3.1

Factor $2$ out of $6$ .

$\frac{2 (3)}{4 x^{2}}$

Step 1.4.3.2

Cancel the common factors.

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Step 1.4.3.2.1

Factor $2$ out of $4 x^{2}$ .

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

Step 1.4.3.2.2

Cancel the common factor.

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

Step 1.4.3.2.3

Rewrite the expression.

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

Step 2

Find the second derivative.

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Step 2.1

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

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

Step 2.2

Apply basic rules of exponents.

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Step 2.2.1

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

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

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, ${(a^{m})}^{n} = a^{m n}$ .

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

Step 2.2.2.2

Multiply $2$ by $- 1$ .

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

Step 2.3

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

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

Step 2.4

Simplify terms.

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Step 2.4.1

Combine $- 2$ and $\frac{3}{2}$ .

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

Step 2.4.2

Multiply $- 2$ by $3$ .

$\frac{- 6}{2} x^{- 3}$

Step 2.4.3

Combine $\frac{- 6}{2}$ and $x^{- 3}$ .

$\frac{- 6 x^{- 3}}{2}$

Step 2.4.4

Move $x^{- 3}$ to the denominator using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$\frac{- 6}{2 x^{3}}$

Step 2.4.5

Cancel the common factor of $- 6$ and $2$ .

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Step 2.4.5.1

Factor $2$ out of $- 6$ .

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

Step 2.4.5.2

Cancel the common factors.

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Step 2.4.5.2.1

Factor $2$ out of $2 x^{3}$ .

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

Step 2.4.5.2.2

Cancel the common factor.

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

Step 2.4.5.2.3

Rewrite the expression.

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

Step 2.4.6

Move the negative in front of the fraction.

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