Calculus Examples

$f (x) = \frac{3 x^{2} + 5 x - 4}{x}$

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 $f (x) = 3 x^{2} + 5 x - 4$ and $g (x) = x$ .

$\frac{x \frac{d}{d x} [3 x^{2} + 5 x - 4] - (3 x^{2} + 5 x - 4) \frac{d}{d x} [x]}{x^{2}}$

Step 1.2

Differentiate.

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Step 1.2.1

By the Sum Rule, the derivative of $3 x^{2} + 5 x - 4$ with respect to $x$ is $\frac{d}{d x} [3 x^{2}] + \frac{d}{d x} [5 x] + \frac{d}{d x} [- 4]$ .

$\frac{x (\frac{d}{d x} [3 x^{2}] + \frac{d}{d x} [5 x] + \frac{d}{d x} [- 4]) - (3 x^{2} + 5 x - 4) \frac{d}{d x} [x]}{x^{2}}$

Step 1.2.2

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

$\frac{x (3 \frac{d}{d x} [x^{2}] + \frac{d}{d x} [5 x] + \frac{d}{d x} [- 4]) - (3 x^{2} + 5 x - 4) \frac{d}{d x} [x]}{x^{2}}$

Step 1.2.3

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

$\frac{x (3 (2 x) + \frac{d}{d x} [5 x] + \frac{d}{d x} [- 4]) - (3 x^{2} + 5 x - 4) \frac{d}{d x} [x]}{x^{2}}$

Step 1.2.4

Multiply $2$ by $3$ .

$\frac{x (6 x + \frac{d}{d x} [5 x] + \frac{d}{d x} [- 4]) - (3 x^{2} + 5 x - 4) \frac{d}{d x} [x]}{x^{2}}$

Step 1.2.5

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

$\frac{x (6 x + 5 \frac{d}{d x} [x] + \frac{d}{d x} [- 4]) - (3 x^{2} + 5 x - 4) \frac{d}{d x} [x]}{x^{2}}$

Step 1.2.6

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

$\frac{x (6 x + 5 \cdot 1 + \frac{d}{d x} [- 4]) - (3 x^{2} + 5 x - 4) \frac{d}{d x} [x]}{x^{2}}$

Step 1.2.7

Multiply $5$ by $1$ .

$\frac{x (6 x + 5 + \frac{d}{d x} [- 4]) - (3 x^{2} + 5 x - 4) \frac{d}{d x} [x]}{x^{2}}$

Step 1.2.8

Since $- 4$ is constant with respect to $x$ , the derivative of $- 4$ with respect to $x$ is $0$ .

$\frac{x (6 x + 5 + 0) - (3 x^{2} + 5 x - 4) \frac{d}{d x} [x]}{x^{2}}$

Step 1.2.9

Add $6 x + 5$ and $0$ .

$\frac{x (6 x + 5) - (3 x^{2} + 5 x - 4) \frac{d}{d x} [x]}{x^{2}}$

Step 1.2.10

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

$\frac{x (6 x + 5) - (3 x^{2} + 5 x - 4) \cdot 1}{x^{2}}$

Step 1.2.11

Multiply $- 1$ by $1$ .

$\frac{x (6 x + 5) - (3 x^{2} + 5 x - 4)}{x^{2}}$

Step 1.3

Simplify.

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Step 1.3.1

Apply the distributive property.

$\frac{x (6 x) + x \cdot 5 - (3 x^{2} + 5 x - 4)}{x^{2}}$

Step 1.3.2

Apply the distributive property.

$\frac{x (6 x) + x \cdot 5 - (3 x^{2}) - (5 x) - - 4}{x^{2}}$

Step 1.3.3

Simplify the numerator.

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Step 1.3.3.1

Simplify each term.

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Step 1.3.3.1.1

Rewrite using the commutative property of multiplication.

$\frac{6 x \cdot x + x \cdot 5 - (3 x^{2}) - (5 x) - - 4}{x^{2}}$

Step 1.3.3.1.2

Multiply $x$ by $x$ by adding the exponents.

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Step 1.3.3.1.2.1

Move $x$ .

$\frac{6 (x \cdot x) + x \cdot 5 - (3 x^{2}) - (5 x) - - 4}{x^{2}}$

Step 1.3.3.1.2.2

Multiply $x$ by $x$ .

$\frac{6 x^{2} + x \cdot 5 - (3 x^{2}) - (5 x) - - 4}{x^{2}}$

Step 1.3.3.1.3

Move $5$ to the left of $x$ .

$\frac{6 x^{2} + 5 \cdot x - (3 x^{2}) - (5 x) - - 4}{x^{2}}$

Step 1.3.3.1.4

Multiply $3$ by $- 1$ .

$\frac{6 x^{2} + 5 x - 3 x^{2} - (5 x) - - 4}{x^{2}}$

Step 1.3.3.1.5

Multiply $5$ by $- 1$ .

$\frac{6 x^{2} + 5 x - 3 x^{2} - 5 x - - 4}{x^{2}}$

Step 1.3.3.1.6

Multiply $- 1$ by $- 4$ .

$\frac{6 x^{2} + 5 x - 3 x^{2} - 5 x + 4}{x^{2}}$

Step 1.3.3.2

Combine the opposite terms in $6 x^{2} + 5 x - 3 x^{2} - 5 x + 4$ .

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Step 1.3.3.2.1

Subtract $5 x$ from $5 x$ .

$\frac{6 x^{2} - 3 x^{2} + 0 + 4}{x^{2}}$

Step 1.3.3.2.2

Add $6 x^{2} - 3 x^{2}$ and $0$ .

$\frac{6 x^{2} - 3 x^{2} + 4}{x^{2}}$

Step 1.3.3.3

Subtract $3 x^{2}$ from $6 x^{2}$ .

$f' (x) = \frac{3 x^{2} + 4}{x^{2}}$

Step 2

Find the second derivative.

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Step 2.1

$\frac{x^{2} \frac{d}{d x} [3 x^{2} + 4] - (3 x^{2} + 4) \frac{d}{d x} [x^{2}]}{{(x^{2})}^{2}}$

Step 2.2

Differentiate.

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Step 2.2.1

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

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Step 2.2.1.1

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

$\frac{x^{2} \frac{d}{d x} [3 x^{2} + 4] - (3 x^{2} + 4) \frac{d}{d x} [x^{2}]}{x^{2 \cdot 2}}$

Step 2.2.1.2

Multiply $2$ by $2$ .

$\frac{x^{2} \frac{d}{d x} [3 x^{2} + 4] - (3 x^{2} + 4) \frac{d}{d x} [x^{2}]}{x^{4}}$

Step 2.2.2

By the Sum Rule, the derivative of $3 x^{2} + 4$ with respect to $x$ is $\frac{d}{d x} [3 x^{2}] + \frac{d}{d x} [4]$ .

$\frac{x^{2} (\frac{d}{d x} [3 x^{2}] + \frac{d}{d x} [4]) - (3 x^{2} + 4) \frac{d}{d x} [x^{2}]}{x^{4}}$

Step 2.2.3

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

$\frac{x^{2} (3 \frac{d}{d x} [x^{2}] + \frac{d}{d x} [4]) - (3 x^{2} + 4) \frac{d}{d x} [x^{2}]}{x^{4}}$

Step 2.2.4

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

$\frac{x^{2} (3 (2 x) + \frac{d}{d x} [4]) - (3 x^{2} + 4) \frac{d}{d x} [x^{2}]}{x^{4}}$

Step 2.2.5

Multiply $2$ by $3$ .

$\frac{x^{2} (6 x + \frac{d}{d x} [4]) - (3 x^{2} + 4) \frac{d}{d x} [x^{2}]}{x^{4}}$

Step 2.2.6

Since $4$ is constant with respect to $x$ , the derivative of $4$ with respect to $x$ is $0$ .

$\frac{x^{2} (6 x + 0) - (3 x^{2} + 4) \frac{d}{d x} [x^{2}]}{x^{4}}$

Step 2.2.7

Add $6 x$ and $0$ .

$\frac{x^{2} (6 x) - (3 x^{2} + 4) \frac{d}{d x} [x^{2}]}{x^{4}}$

Step 2.3

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

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Step 2.3.1

Move $x$ .

$\frac{x \cdot x^{2} \cdot 6 - (3 x^{2} + 4) \frac{d}{d x} [x^{2}]}{x^{4}}$

Step 2.3.2

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

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Step 2.3.2.1

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

$\frac{x^{1} x^{2} \cdot 6 - (3 x^{2} + 4) \frac{d}{d x} [x^{2}]}{x^{4}}$

Step 2.3.2.2

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

$\frac{x^{1 + 2} \cdot 6 - (3 x^{2} + 4) \frac{d}{d x} [x^{2}]}{x^{4}}$

Step 2.3.3

Add $1$ and $2$ .

$\frac{x^{3} \cdot 6 - (3 x^{2} + 4) \frac{d}{d x} [x^{2}]}{x^{4}}$

Step 2.4

Move $6$ to the left of $x^{3}$ .

$\frac{6 \cdot x^{3} - (3 x^{2} + 4) \frac{d}{d x} [x^{2}]}{x^{4}}$

Step 2.5

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

$\frac{6 x^{3} - (3 x^{2} + 4) (2 x)}{x^{4}}$

Step 2.6

Multiply $2$ by $- 1$ .

$\frac{6 x^{3} - 2 (3 x^{2} + 4) x}{x^{4}}$

Step 2.7

Simplify.

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Step 2.7.1

Apply the distributive property.

$\frac{6 x^{3} + (- 2 (3 x^{2}) - 2 \cdot 4) x}{x^{4}}$

Step 2.7.2

Apply the distributive property.

$\frac{6 x^{3} - 2 (3 x^{2}) x - 2 \cdot 4 x}{x^{4}}$

Step 2.7.3

Simplify the numerator.

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Step 2.7.3.1

Simplify each term.

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Step 2.7.3.1.1

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

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Step 2.7.3.1.1.1

Move $x$ .

$\frac{6 x^{3} - 2 (3 (x \cdot x^{2})) - 2 \cdot 4 x}{x^{4}}$

Step 2.7.3.1.1.2

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

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Step 2.7.3.1.1.2.1

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

$\frac{6 x^{3} - 2 (3 (x^{1} x^{2})) - 2 \cdot 4 x}{x^{4}}$

Step 2.7.3.1.1.2.2

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

$\frac{6 x^{3} - 2 (3 x^{1 + 2}) - 2 \cdot 4 x}{x^{4}}$

Step 2.7.3.1.1.3

Add $1$ and $2$ .

$\frac{6 x^{3} - 2 (3 x^{3}) - 2 \cdot 4 x}{x^{4}}$

Step 2.7.3.1.2

Multiply $3$ by $- 2$ .

$\frac{6 x^{3} - 6 x^{3} - 2 \cdot 4 x}{x^{4}}$

Step 2.7.3.1.3

Multiply $- 2$ by $4$ .

$\frac{6 x^{3} - 6 x^{3} - 8 x}{x^{4}}$

Step 2.7.3.2

Subtract $6 x^{3}$ from $6 x^{3}$ .

$\frac{0 - 8 x}{x^{4}}$

Step 2.7.3.3

Subtract $8 x$ from $0$ .

$\frac{- 8 x}{x^{4}}$

Step 2.7.4

Combine terms.

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Step 2.7.4.1

Cancel the common factor of $x$ and $x^{4}$ .

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Step 2.7.4.1.1

Factor $x$ out of $- 8 x$ .

$\frac{x \cdot - 8}{x^{4}}$

Step 2.7.4.1.2

Cancel the common factors.

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Step 2.7.4.1.2.1

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

$\frac{x \cdot - 8}{x \cdot x^{3}}$

Step 2.7.4.1.2.2

Cancel the common factor.

$\frac{x \cdot - 8}{x \cdot x^{3}}$

Step 2.7.4.1.2.3

Rewrite the expression.

$\frac{- 8}{x^{3}}$

Step 2.7.4.2

Move the negative in front of the fraction.

$f'' (x) = - \frac{8}{x^{3}}$

Step 3

The second derivative of $f (x)$ with respect to $x$ is $- \frac{8}{x^{3}}$ .

$- \frac{8}{x^{3}}$