Calculus Examples

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

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

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

Step 1.2

Differentiate.

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Step 1.2.1

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

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

Step 1.2.2

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

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

Step 1.2.3

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

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

Step 1.2.4

Multiply $3$ by $1$ .

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

Step 1.2.5

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

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

Step 1.2.6

Simplify the expression.

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Step 1.2.6.1

Add $3$ and $0$ .

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

Step 1.2.6.2

Move $3$ to the left of $2 x - 3$ .

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

Step 1.2.7

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

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

Step 1.2.8

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

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

Step 1.2.9

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

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

Step 1.2.10

Multiply $2$ by $1$ .

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

Step 1.2.11

Since $- 3$ is constant with respect to $x$ , the derivative of $- 3$ with respect to $x$ is $0$ .

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

Step 1.2.12

Simplify the expression.

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Step 1.2.12.1

Add $2$ and $0$ .

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

Step 1.2.12.2

Multiply $2$ by $- 1$ .

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

Step 1.3

Simplify.

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Step 1.3.1

Apply the distributive property.

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

Step 1.3.2

Apply the distributive property.

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

Step 1.3.3

Simplify the numerator.

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Step 1.3.3.1

Combine the opposite terms in $3 (2 x) + 3 \cdot - 3 - 2 (3 x) - 2 \cdot 4$ .

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Step 1.3.3.1.1

Reorder the factors in the terms $3 (2 x)$ and $- 2 (3 x)$ .

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

Step 1.3.3.1.2

Subtract $2 \cdot 3 x$ from $2 \cdot 3 x$ .

$\frac{3 \cdot - 3 + 0 - 2 \cdot 4}{{(2 x - 3)}^{2}}$

Step 1.3.3.1.3

Add $3 \cdot - 3$ and $0$ .

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

Step 1.3.3.2

Simplify each term.

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Step 1.3.3.2.1

Multiply $3$ by $- 3$ .

$\frac{- 9 - 2 \cdot 4}{{(2 x - 3)}^{2}}$

Step 1.3.3.2.2

Multiply $- 2$ by $4$ .

$\frac{- 9 - 8}{{(2 x - 3)}^{2}}$

Step 1.3.3.3

Subtract $8$ from $- 9$ .

$\frac{- 17}{{(2 x - 3)}^{2}}$

Step 1.3.4

Move the negative in front of the fraction.

$f' (x) = - \frac{17}{{(2 x - 3)}^{2}}$

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 $- 17$ is constant with respect to $x$ , the derivative of $- \frac{17}{{(2 x - 3)}^{2}}$ with respect to $x$ is $- 17 \frac{d}{d x} [\frac{1}{{(2 x - 3)}^{2}}]$ .

$- 17 \frac{d}{d x} [\frac{1}{{(2 x - 3)}^{2}}]$

Step 2.1.2

Apply basic rules of exponents.

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Step 2.1.2.1

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

$- 17 \frac{d}{d x} [{({(2 x - 3)}^{2})}^{- 1}]$

Step 2.1.2.2

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

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Step 2.1.2.2.1

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

$- 17 \frac{d}{d x} [{(2 x - 3)}^{2 \cdot - 1}]$

Step 2.1.2.2.2

Multiply $2$ by $- 1$ .

$- 17 \frac{d}{d x} [{(2 x - 3)}^{- 2}]$

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 $g (x) = 2 x - 3$ .

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Step 2.2.1

To apply the Chain Rule, set $u$ as $2 x - 3$ .

$- 17 (\frac{d}{d u} [u^{- 2}] \frac{d}{d x} [2 x - 3])$

Step 2.2.2

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

$- 17 (- 2 u^{- 3} \frac{d}{d x} [2 x - 3])$

Step 2.2.3

Replace all occurrences of $u$ with $2 x - 3$ .

$- 17 (- 2 {(2 x - 3)}^{- 3} \frac{d}{d x} [2 x - 3])$

Step 2.3

Differentiate.

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Step 2.3.1

Multiply $- 2$ by $- 17$ .

$34 ({(2 x - 3)}^{- 3} \frac{d}{d x} [2 x - 3])$

Step 2.3.2

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

$34 {(2 x - 3)}^{- 3} (\frac{d}{d x} [2 x] + \frac{d}{d x} [- 3])$

Step 2.3.3

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

$34 {(2 x - 3)}^{- 3} (2 \frac{d}{d x} [x] + \frac{d}{d x} [- 3])$

Step 2.3.4

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

$34 {(2 x - 3)}^{- 3} (2 \cdot 1 + \frac{d}{d x} [- 3])$

Step 2.3.5

Multiply $2$ by $1$ .

$34 {(2 x - 3)}^{- 3} (2 + \frac{d}{d x} [- 3])$

Step 2.3.6

Since $- 3$ is constant with respect to $x$ , the derivative of $- 3$ with respect to $x$ is $0$ .

$34 {(2 x - 3)}^{- 3} (2 + 0)$

Step 2.3.7

Simplify the expression.

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Step 2.3.7.1

Add $2$ and $0$ .

$34 {(2 x - 3)}^{- 3} \cdot 2$

Step 2.3.7.2

Multiply $2$ by $34$ .

$68 {(2 x - 3)}^{- 3}$

Step 2.4

Simplify.

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Step 2.4.1

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

$68 \frac{1}{{(2 x - 3)}^{3}}$

Step 2.4.2

Combine $68$ and $\frac{1}{{(2 x - 3)}^{3}}$ .

$f'' (x) = \frac{68}{{(2 x - 3)}^{3}}$