Calculus Examples

$y = \frac{2 x - 1}{3 x + 4}$

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 $f (x) = 2 x - 1$ and $g (x) = 3 x + 4$ .

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

Step 1.2

Differentiate.

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Step 1.2.1

By the Sum Rule, the derivative of $2 x - 1$ with respect to $x$ is $\frac{d}{d x} [2 x] + \frac{d}{d x} [- 1]$ .

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

Step 1.2.2

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

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

Step 1.2.3

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

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

Step 1.2.4

Multiply $2$ by $1$ .

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

Step 1.2.5

Since $- 1$ is constant with respect to $x$ , the derivative of $- 1$ with respect to $x$ is $0$ .

$f' (x) = \frac{(3 x + 4) (2 + 0) - (2 x - 1) \frac{d}{d x} (3 x + 4)}{{(3 x + 4)}^{2}}$

Step 1.2.6

Simplify the expression.

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Step 1.2.6.1

Add $2$ and $0$ .

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

Step 1.2.6.2

Move $2$ to the left of $3 x + 4$ .

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

Step 1.2.7

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

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

Step 1.2.8

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

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

Step 1.2.9

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

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

Step 1.2.10

Multiply $3$ by $1$ .

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

Step 1.2.11

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

$f' (x) = \frac{2 (3 x + 4) - (2 x - 1) (3 + 0)}{{(3 x + 4)}^{2}}$

Step 1.2.12

Simplify the expression.

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Step 1.2.12.1

Add $3$ and $0$ .

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

Step 1.2.12.2

Multiply $3$ by $- 1$ .

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

Step 1.3

Simplify.

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Step 1.3.1

Apply the distributive property.

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

Step 1.3.2

Apply the distributive property.

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

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

Multiply $3$ by $2$ .

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

Step 1.3.3.1.2

Multiply $2$ by $4$ .

$f' (x) = \frac{6 x + 8 - 3 (2 x) - 3 \cdot - 1}{{(3 x + 4)}^{2}}$

Step 1.3.3.1.3

Multiply $2$ by $- 3$ .

$f' (x) = \frac{6 x + 8 - 6 x - 3 \cdot - 1}{{(3 x + 4)}^{2}}$

Step 1.3.3.1.4

Multiply $- 3$ by $- 1$ .

$f' (x) = \frac{6 x + 8 - 6 x + 3}{{(3 x + 4)}^{2}}$

Step 1.3.3.2

Combine the opposite terms in $6 x + 8 - 6 x + 3$ .

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Step 1.3.3.2.1

Subtract $6 x$ from $6 x$ .

$f' (x) = \frac{0 + 8 + 3}{{(3 x + 4)}^{2}}$

Step 1.3.3.2.2

Add $0$ and $8$ .

$f' (x) = \frac{8 + 3}{{(3 x + 4)}^{2}}$

Step 1.3.3.3

Add $8$ and $3$ .

$f' (x) = \frac{11}{{(3 x + 4)}^{2}}$

Step 2

Find the second derivative.

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Step 2.1

Differentiate using the Constant Multiple Rule.

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Step 2.1.1

Since $11$ is constant with respect to $x$ , the derivative of $\frac{11}{{(3 x + 4)}^{2}}$ with respect to $x$ is $11 \frac{d}{d x} [\frac{1}{{(3 x + 4)}^{2}}]$ .

$f'' (x) = 11 \frac{d}{d x} (\frac{1}{{(3 x + 4)}^{2}})$

Step 2.1.2

Apply basic rules of exponents.

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Step 2.1.2.1

Rewrite $\frac{1}{{(3 x + 4)}^{2}}$ as ${({(3 x + 4)}^{2})}^{- 1}$ .

$f'' (x) = 11 \frac{d}{d x} ({({(3 x + 4)}^{2})}^{- 1})$

Step 2.1.2.2

Multiply the exponents in ${({(3 x + 4)}^{2})}^{- 1}$ .

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Step 2.1.2.2.1

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

$f'' (x) = 11 \frac{d}{d x} ({(3 x + 4)}^{2 \cdot - 1})$

Step 2.1.2.2.2

Multiply $2$ by $- 1$ .

$f'' (x) = 11 \frac{d}{d x} ({(3 x + 4)}^{- 2})$

Step 2.2

Differentiate using the chain rule, which states that $\frac{d}{d x} [f (g (x))]$ is $f' (g (x)) g' (x)$ where $f (x) = x^{- 2}$ and $g (x) = 3 x + 4$ .

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Step 2.2.1

To apply the Chain Rule, set $u$ as $3 x + 4$ .

$f'' (x) = 11 (\frac{d}{d u} (u^{- 2}) \frac{d}{d x} (3 x + 4))$

Step 2.2.2

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

$f'' (x) = 11 (- 2 u^{- 3} \frac{d}{d x} (3 x + 4))$

Step 2.2.3

Replace all occurrences of $u$ with $3 x + 4$ .

$f'' (x) = 11 (- 2 {(3 x + 4)}^{- 3} \frac{d}{d x} (3 x + 4))$

Step 2.3

Differentiate.

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Step 2.3.1

Multiply $- 2$ by $11$ .

$f'' (x) = - 22 ({(3 x + 4)}^{- 3} \frac{d}{d x} (3 x + 4))$

Step 2.3.2

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

$f'' (x) = - 22 {(3 x + 4)}^{- 3} (\frac{d}{d x} (3 x) + \frac{d}{d x} (4))$

Step 2.3.3

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

$f'' (x) = - 22 {(3 x + 4)}^{- 3} (3 \frac{d}{d x} (x) + \frac{d}{d x} (4))$

Step 2.3.4

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

$f'' (x) = - 22 {(3 x + 4)}^{- 3} (3 \cdot 1 + \frac{d}{d x} (4))$

Step 2.3.5

Multiply $3$ by $1$ .

$f'' (x) = - 22 {(3 x + 4)}^{- 3} (3 + \frac{d}{d x} (4))$

Step 2.3.6

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

$f'' (x) = - 22 {(3 x + 4)}^{- 3} (3 + 0)$

Step 2.3.7

Simplify the expression.

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Step 2.3.7.1

Add $3$ and $0$ .

$f'' (x) = - 22 {(3 x + 4)}^{- 3} \cdot 3$

Step 2.3.7.2

Multiply $3$ by $- 22$ .

$f'' (x) = - 66 {(3 x + 4)}^{- 3}$

Step 2.4

Simplify.

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Step 2.4.1

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

$f'' (x) = - 66 \frac{1}{{(3 x + 4)}^{3}}$

Step 2.4.2

Combine terms.

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Step 2.4.2.1

Combine $- 66$ and $\frac{1}{{(3 x + 4)}^{3}}$ .

$f'' (x) = \frac{- 66}{{(3 x + 4)}^{3}}$

Step 2.4.2.2

Move the negative in front of the fraction.

$f'' (x) = - \frac{66}{{(3 x + 4)}^{3}}$

Step 3

Find the third derivative.

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Step 3.1

Differentiate using the Constant Multiple Rule.

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Step 3.1.1

Since $- 66$ is constant with respect to $x$ , the derivative of $- \frac{66}{{(3 x + 4)}^{3}}$ with respect to $x$ is $- 66 \frac{d}{d x} [\frac{1}{{(3 x + 4)}^{3}}]$ .

$f''' (x) = - 66 \frac{d}{d x} \frac{1}{{(3 x + 4)}^{3}}$

Step 3.1.2

Apply basic rules of exponents.

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Step 3.1.2.1

Rewrite $\frac{1}{{(3 x + 4)}^{3}}$ as ${({(3 x + 4)}^{3})}^{- 1}$ .

$f''' (x) = - 66 \frac{d}{d x} {({(3 x + 4)}^{3})}^{- 1}$

Step 3.1.2.2

Multiply the exponents in ${({(3 x + 4)}^{3})}^{- 1}$ .

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Step 3.1.2.2.1

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

$f''' (x) = - 66 \frac{d}{d x} {(3 x + 4)}^{3 \cdot - 1}$

Step 3.1.2.2.2

Multiply $3$ by $- 1$ .

$f''' (x) = - 66 \frac{d}{d x} {(3 x + 4)}^{- 3}$

Step 3.2

Differentiate using the chain rule, which states that $\frac{d}{d x} [f (g (x))]$ is $f' (g (x)) g' (x)$ where $f (x) = x^{- 3}$ and $g (x) = 3 x + 4$ .

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Step 3.2.1

To apply the Chain Rule, set $u$ as $3 x + 4$ .

$f''' (x) = - 66 (\frac{d}{d u} (u^{- 3}) \frac{d}{d x} (3 x + 4))$

Step 3.2.2

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

$f''' (x) = - 66 (- 3 u^{- 4} \frac{d}{d x} (3 x + 4))$

Step 3.2.3

Replace all occurrences of $u$ with $3 x + 4$ .

$f''' (x) = - 66 (- 3 {(3 x + 4)}^{- 4} \frac{d}{d x} (3 x + 4))$

Step 3.3

Differentiate.

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Step 3.3.1

Multiply $- 3$ by $- 66$ .

$f''' (x) = 198 ({(3 x + 4)}^{- 4} \frac{d}{d x} (3 x + 4))$

Step 3.3.2

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

$f''' (x) = 198 {(3 x + 4)}^{- 4} (\frac{d}{d x} (3 x) + \frac{d}{d x} (4))$

Step 3.3.3

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

$f''' (x) = 198 {(3 x + 4)}^{- 4} (3 \frac{d}{d x} (x) + \frac{d}{d x} (4))$

Step 3.3.4

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

$f''' (x) = 198 {(3 x + 4)}^{- 4} (3 \cdot 1 + \frac{d}{d x} (4))$

Step 3.3.5

Multiply $3$ by $1$ .

$f''' (x) = 198 {(3 x + 4)}^{- 4} (3 + \frac{d}{d x} (4))$

Step 3.3.6

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

$f''' (x) = 198 {(3 x + 4)}^{- 4} (3 + 0)$

Step 3.3.7

Simplify the expression.

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Step 3.3.7.1

Add $3$ and $0$ .

$f''' (x) = 198 {(3 x + 4)}^{- 4} \cdot 3$

Step 3.3.7.2

Multiply $3$ by $198$ .

$f''' (x) = 594 {(3 x + 4)}^{- 4}$

Step 3.4

Simplify.

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Step 3.4.1

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

$f''' (x) = 594 (\frac{1}{{(3 x + 4)}^{4}})$

Step 3.4.2

Combine $594$ and $\frac{1}{{(3 x + 4)}^{4}}$ .

$f''' (x) = \frac{594}{{(3 x + 4)}^{4}}$

Step 4

Find the fourth derivative.

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Step 4.1

Differentiate using the Constant Multiple Rule.

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Step 4.1.1

Since $594$ is constant with respect to $x$ , the derivative of $\frac{594}{{(3 x + 4)}^{4}}$ with respect to $x$ is $594 \frac{d}{d x} [\frac{1}{{(3 x + 4)}^{4}}]$ .

$f^{4} (x) = 594 \frac{d}{d x} (\frac{1}{{(3 x + 4)}^{4}})$

Step 4.1.2

Apply basic rules of exponents.

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Step 4.1.2.1

Rewrite $\frac{1}{{(3 x + 4)}^{4}}$ as ${({(3 x + 4)}^{4})}^{- 1}$ .

$f^{4} (x) = 594 \frac{d}{d x} ({({(3 x + 4)}^{4})}^{- 1})$

Step 4.1.2.2

Multiply the exponents in ${({(3 x + 4)}^{4})}^{- 1}$ .

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Step 4.1.2.2.1

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

$f^{4} (x) = 594 \frac{d}{d x} ({(3 x + 4)}^{4 \cdot - 1})$

Step 4.1.2.2.2

Multiply $4$ by $- 1$ .

$f^{4} (x) = 594 \frac{d}{d x} ({(3 x + 4)}^{- 4})$

Step 4.2

Differentiate using the chain rule, which states that $\frac{d}{d x} [f (g (x))]$ is $f' (g (x)) g' (x)$ where $f (x) = x^{- 4}$ and $g (x) = 3 x + 4$ .

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Step 4.2.1

To apply the Chain Rule, set $u$ as $3 x + 4$ .

$f^{4} (x) = 594 (\frac{d}{d u} (u^{- 4}) \frac{d}{d x} (3 x + 4))$

Step 4.2.2

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

$f^{4} (x) = 594 (- 4 u^{- 5} \frac{d}{d x} (3 x + 4))$

Step 4.2.3

Replace all occurrences of $u$ with $3 x + 4$ .

$f^{4} (x) = 594 (- 4 {(3 x + 4)}^{- 5} \frac{d}{d x} (3 x + 4))$

Step 4.3

Differentiate.

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Step 4.3.1

Multiply $- 4$ by $594$ .

$f^{4} (x) = - 2376 ({(3 x + 4)}^{- 5} \frac{d}{d x} (3 x + 4))$

Step 4.3.2

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

$f^{4} (x) = - 2376 {(3 x + 4)}^{- 5} (\frac{d}{d x} (3 x) + \frac{d}{d x} (4))$

Step 4.3.3

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

$f^{4} (x) = - 2376 {(3 x + 4)}^{- 5} (3 \frac{d}{d x} (x) + \frac{d}{d x} (4))$

Step 4.3.4

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

$f^{4} (x) = - 2376 {(3 x + 4)}^{- 5} (3 \cdot 1 + \frac{d}{d x} (4))$

Step 4.3.5

Multiply $3$ by $1$ .

$f^{4} (x) = - 2376 {(3 x + 4)}^{- 5} (3 + \frac{d}{d x} (4))$

Step 4.3.6

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

$f^{4} (x) = - 2376 {(3 x + 4)}^{- 5} (3 + 0)$

Step 4.3.7

Simplify the expression.

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Step 4.3.7.1

Add $3$ and $0$ .

$f^{4} (x) = - 2376 {(3 x + 4)}^{- 5} \cdot 3$

Step 4.3.7.2

Multiply $3$ by $- 2376$ .

$f^{4} (x) = - 7128 {(3 x + 4)}^{- 5}$

Step 4.4

Simplify.

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Step 4.4.1

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

$f^{4} (x) = - 7128 \frac{1}{{(3 x + 4)}^{5}}$

Step 4.4.2

Combine terms.

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Step 4.4.2.1

Combine $- 7128$ and $\frac{1}{{(3 x + 4)}^{5}}$ .

$f^{4} (x) = \frac{- 7128}{{(3 x + 4)}^{5}}$

Step 4.4.2.2

Move the negative in front of the fraction.

$f^{4} (x) = - \frac{7128}{{(3 x + 4)}^{5}}$