Calculus Examples

$y = \frac{3 x - 5}{x}$

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 $f (x) = 3 x - 5$ and $g (x) = x$ .

$\frac{x \frac{d}{d x} [3 x - 5] - (3 x - 5) \frac{d}{d x} [x]}{x^{2}}$

Step 1.2

Differentiate.

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Step 1.2.1

By the Sum Rule, the derivative of $3 x - 5$ with respect to $x$ is $\frac{d}{d x} [3 x] + \frac{d}{d x} [- 5]$ .

$\frac{x (\frac{d}{d x} [3 x] + \frac{d}{d x} [- 5]) - (3 x - 5) \frac{d}{d x} [x]}{x^{2}}$

Step 1.2.2

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

$\frac{x (3 \frac{d}{d x} [x] + \frac{d}{d x} [- 5]) - (3 x - 5) \frac{d}{d x} [x]}{x^{2}}$

Step 1.2.3

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

$\frac{x (3 \cdot 1 + \frac{d}{d x} [- 5]) - (3 x - 5) \frac{d}{d x} [x]}{x^{2}}$

Step 1.2.4

Multiply $3$ by $1$ .

$\frac{x (3 + \frac{d}{d x} [- 5]) - (3 x - 5) \frac{d}{d x} [x]}{x^{2}}$

Step 1.2.5

Since $- 5$ is constant with respect to $x$ , the derivative of $- 5$ with respect to $x$ is $0$ .

$\frac{x (3 + 0) - (3 x - 5) \frac{d}{d x} [x]}{x^{2}}$

Step 1.2.6

Simplify the expression.

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Step 1.2.6.1

Add $3$ and $0$ .

$\frac{x \cdot 3 - (3 x - 5) \frac{d}{d x} [x]}{x^{2}}$

Step 1.2.6.2

Move $3$ to the left of $x$ .

$\frac{3 \cdot x - (3 x - 5) \frac{d}{d x} [x]}{x^{2}}$

Step 1.2.7

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

$\frac{3 x - (3 x - 5) \cdot 1}{x^{2}}$

Step 1.2.8

Multiply $- 1$ by $1$ .

$\frac{3 x - (3 x - 5)}{x^{2}}$

Step 1.3

Simplify.

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Step 1.3.1

Apply the distributive property.

$\frac{3 x - (3 x) - - 5}{x^{2}}$

Step 1.3.2

Simplify the numerator.

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Step 1.3.2.1

Simplify each term.

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Step 1.3.2.1.1

Multiply $3$ by $- 1$ .

$\frac{3 x - 3 x - - 5}{x^{2}}$

Step 1.3.2.1.2

Multiply $- 1$ by $- 5$ .

$\frac{3 x - 3 x + 5}{x^{2}}$

Step 1.3.2.2

Subtract $3 x$ from $3 x$ .

$\frac{0 + 5}{x^{2}}$

Step 1.3.2.3

Add $0$ and $5$ .

$f' (x) = \frac{5}{x^{2}}$

Step 2

Find the second derivative.

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Step 2.1

Since $5$ is constant with respect to $x$ , the derivative of $\frac{5}{x^{2}}$ with respect to $x$ is $5 \frac{d}{d x} [\frac{1}{x^{2}}]$ .

$5 \frac{d}{d x} [\frac{1}{x^{2}}]$

Step 2.2

Apply basic rules of exponents.

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Step 2.2.1

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

$5 \frac{d}{d x} [{(x^{2})}^{- 1}]$

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

$5 \frac{d}{d x} [x^{2 \cdot - 1}]$

Step 2.2.2.2

Multiply $2$ by $- 1$ .

$5 \frac{d}{d x} [x^{- 2}]$

Step 2.3

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

$5 (- 2 x^{- 3})$

Step 2.4

Multiply $- 2$ by $5$ .

$- 10 x^{- 3}$

Step 2.5

Simplify.

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Step 2.5.1

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

$- 10 \frac{1}{x^{3}}$

Step 2.5.2

Combine terms.

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Step 2.5.2.1

Combine $- 10$ and $\frac{1}{x^{3}}$ .

$\frac{- 10}{x^{3}}$

Step 2.5.2.2

Move the negative in front of the fraction.

$f'' (x) = - \frac{10}{x^{3}}$