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 $g (x) = x$ .

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

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

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

Step 1.2.2

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

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

Step 1.2.3

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

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

Step 1.2.4

Multiply $2$ by $3$ .

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

Step 1.2.5

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

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

Step 1.2.6

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

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

Step 1.2.7

Multiply $5$ by $1$ .

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

Step 1.2.8

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

$f' (x) = \frac{x (6 x + 5 + 0) - (3 x^{2} + 5 x - 4) \frac{d}{d x} x}{x^{2}}$

Step 1.2.9

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

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

Step 1.2.10

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

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

Step 1.2.11

Multiply $- 1$ by $1$ .

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

Step 1.3

Simplify.

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Step 1.3.1

Apply the distributive property.

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

Step 1.3.2

Apply the distributive property.

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

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.

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

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 $x$ .

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

Step 1.3.3.1.2.2

Multiply $x$ by $x$ .

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

Step 1.3.3.1.3

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

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

Step 1.3.3.1.4

Multiply $3$ by $- 1$ .

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

Step 1.3.3.1.5

Multiply $5$ by $- 1$ .

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

Step 1.3.3.1.6

Multiply $- 1$ by $- 4$ .

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

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 $5 x$ .

$f' (x) = \frac{6 x^{2} - 3 x^{2} + 0 + 4}{x^{2}}$

Step 1.3.3.2.2

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

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

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

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

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

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

Step 2.2.1.2

Multiply $2$ by $2$ .

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

Step 2.2.2

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

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

Step 2.2.3

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

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

Step 2.2.4

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

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

Step 2.2.5

Multiply $2$ by $3$ .

$f'' (x) = \frac{x^{2} (6 x + \frac{d}{d x} (4)) - (3 x^{2} + 4) \frac{d}{d x} x^{2}}{x^{4}}$

Step 2.2.6

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

$f'' (x) = \frac{x^{2} (6 x + 0) - (3 x^{2} + 4) \frac{d}{d x} x^{2}}{x^{4}}$

Step 2.2.7

Add $6 x$ and $0$ .

$f'' (x) = \frac{x^{2} (6 x) - (3 x^{2} + 4) \frac{d}{d x} x^{2}}{x^{4}}$

Step 2.3

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

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Step 2.3.1

Move $x$ .

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

Step 2.3.2

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

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Step 2.3.2.1

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

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

Step 2.3.2.2

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

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

Step 2.3.3

Add $1$ and $2$ .

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

Step 2.4

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

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

Step 2.5

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

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

Step 2.6

Multiply $2$ by $- 1$ .

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

Step 2.7

Simplify.

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Step 2.7.1

Apply the distributive property.

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

Step 2.7.2

Apply the distributive property.

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

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 $x$ .

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

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 $1$ .

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

Step 2.7.3.1.1.2.2

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

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

Step 2.7.3.1.1.3

Add $1$ and $2$ .

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

Step 2.7.3.1.2

Multiply $3$ by $- 2$ .

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

Step 2.7.3.1.3

Multiply $- 2$ by $4$ .

$f'' (x) = \frac{6 x^{3} - 6 x^{3} - 8 x}{x^{4}}$

Step 2.7.3.2

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

$f'' (x) = \frac{0 - 8 x}{x^{4}}$

Step 2.7.3.3

Subtract $8 x$ from $0$ .

$f'' (x) = \frac{- 8 x}{x^{4}}$

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 $- 8 x$ .

$f'' (x) = \frac{x \cdot - 8}{x^{4}}$

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 $x^{4}$ .

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

Step 2.7.4.1.2.2

Cancel the common factor.

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

Step 2.7.4.1.2.3

Rewrite the expression.

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

Step 2.7.4.2

Move the negative in front of the fraction.

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

Step 3

Find the third derivative.

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Step 3.1

Since $- 8$ is constant with respect to $x$ , the derivative of $- \frac{8}{x^{3}}$ with respect to $x$ is $- 8 \frac{d}{d x} [\frac{1}{x^{3}}]$ .

$f''' (x) = - 8 \frac{d}{d x} \frac{1}{x^{3}}$

Step 3.2

Apply basic rules of exponents.

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Step 3.2.1

Rewrite $\frac{1}{x^{3}}$ as ${(x^{3})}^{- 1}$ .

$f''' (x) = - 8 \frac{d}{d x} {(x^{3})}^{- 1}$

Step 3.2.2

Multiply the exponents in ${(x^{3})}^{- 1}$ .

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Step 3.2.2.1

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

$f''' (x) = - 8 \frac{d}{d x} x^{3 \cdot - 1}$

Step 3.2.2.2

Multiply $3$ by $- 1$ .

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

Step 3.3

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = - 3$ .

$f''' (x) = - 8 (- 3 x^{- 4})$

Step 3.4

Multiply $- 3$ by $- 8$ .

$f''' (x) = 24 x^{- 4}$

Step 3.5

Simplify.

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Step 3.5.1

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

$f''' (x) = 24 (\frac{1}{x^{4}})$

Step 3.5.2

Combine $24$ and $\frac{1}{x^{4}}$ .

$f''' (x) = \frac{24}{x^{4}}$

Step 4

Find the fourth derivative.

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Step 4.1

Since $24$ is constant with respect to $x$ , the derivative of $\frac{24}{x^{4}}$ with respect to $x$ is $24 \frac{d}{d x} [\frac{1}{x^{4}}]$ .

$f^{4} (x) = 24 \frac{d}{d x} (\frac{1}{x^{4}})$

Step 4.2

Apply basic rules of exponents.

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Step 4.2.1

Rewrite $\frac{1}{x^{4}}$ as ${(x^{4})}^{- 1}$ .

$f^{4} (x) = 24 \frac{d}{d x} ({(x^{4})}^{- 1})$

Step 4.2.2

Multiply the exponents in ${(x^{4})}^{- 1}$ .

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Step 4.2.2.1

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

$f^{4} (x) = 24 \frac{d}{d x} (x^{4 \cdot - 1})$

Step 4.2.2.2

Multiply $4$ by $- 1$ .

$f^{4} (x) = 24 \frac{d}{d x} (x^{- 4})$

Step 4.3

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = - 4$ .

$f^{4} (x) = 24 (- 4 x^{- 5})$

Step 4.4

Multiply $- 4$ by $24$ .

$f^{4} (x) = - 96 x^{- 5}$

Step 4.5

Simplify.

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Step 4.5.1

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

$f^{4} (x) = - 96 \frac{1}{x^{5}}$

Step 4.5.2

Combine terms.

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Step 4.5.2.1

Combine $- 96$ and $\frac{1}{x^{5}}$ .

$f^{4} (x) = \frac{- 96}{x^{5}}$

Step 4.5.2.2

Move the negative in front of the fraction.

$f^{4} (x) = - \frac{96}{x^{5}}$

Step 5

The fourth derivative of $f (x)$ with respect to $x$ is $- \frac{96}{x^{5}}$ .

$- \frac{96}{x^{5}}$