Calculus Examples

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

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

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

Step 1.2

Differentiate.

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Step 1.2.1

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

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

Step 1.2.2

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

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

Step 1.2.3

Since $2$ is constant with respect to $x$ , the derivative of $2$ with respect to $x$ is $0$ .

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

Step 1.2.4

Simplify the expression.

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Step 1.2.4.1

Add $1$ and $0$ .

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

Step 1.2.4.2

Multiply $x - 3$ by $1$ .

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

Step 1.2.5

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

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

Step 1.2.6

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

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

Step 1.2.7

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

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

Step 1.2.8

Simplify the expression.

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Step 1.2.8.1

Add $1$ and $0$ .

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

Step 1.2.8.2

Multiply $- 1$ by $1$ .

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

Step 1.3

Simplify.

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Step 1.3.1

Apply the distributive property.

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

Step 1.3.2

Simplify the numerator.

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Step 1.3.2.1

Combine the opposite terms in $x - 3 - x - 1 \cdot 2$ .

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Step 1.3.2.1.1

Subtract $x$ from $x$ .

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

Step 1.3.2.1.2

Subtract $3$ from $0$ .

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

Step 1.3.2.2

Multiply $- 1$ by $2$ .

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

Step 1.3.2.3

Subtract $2$ from $- 3$ .

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

Step 1.3.3

Move the negative in front of the fraction.

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

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 $- 5$ is constant with respect to $x$ , the derivative of $- \frac{5}{{(x - 3)}^{2}}$ with respect to $x$ is $- 5 \frac{d}{d x} [\frac{1}{{(x - 3)}^{2}}]$ .

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

Step 2.1.2

Apply basic rules of exponents.

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Step 2.1.2.1

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

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

Step 2.1.2.2

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

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Step 2.1.2.2.1

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

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

Step 2.1.2.2.2

Multiply $2$ by $- 1$ .

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

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 $g (x) = x - 3$ .

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Step 2.2.1

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

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

Step 2.2.2

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

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

Step 2.2.3

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

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

Step 2.3

Differentiate.

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Step 2.3.1

Multiply $- 2$ by $- 5$ .

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

Step 2.3.2

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

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

Step 2.3.3

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

$f'' (x) = 10 {(x - 3)}^{- 3} (1 + \frac{d}{d x} (- 3))$

Step 2.3.4

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

$f'' (x) = 10 {(x - 3)}^{- 3} (1 + 0)$

Step 2.3.5

Simplify the expression.

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Step 2.3.5.1

Add $1$ and $0$ .

$f'' (x) = 10 {(x - 3)}^{- 3} \cdot 1$

Step 2.3.5.2

Multiply $10$ by $1$ .

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

Step 2.4

Simplify.

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Step 2.4.1

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

$f'' (x) = 10 (\frac{1}{{(x - 3)}^{3}})$

Step 2.4.2

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

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

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 $10$ is constant with respect to $x$ , the derivative of $\frac{10}{{(x - 3)}^{3}}$ with respect to $x$ is $10 \frac{d}{d x} [\frac{1}{{(x - 3)}^{3}}]$ .

$f''' (x) = 10 \frac{d}{d x} (\frac{1}{{(x - 3)}^{3}})$

Step 3.1.2

Apply basic rules of exponents.

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Step 3.1.2.1

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

$f''' (x) = 10 \frac{d}{d x} ({({(x - 3)}^{3})}^{- 1})$

Step 3.1.2.2

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

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Step 3.1.2.2.1

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

$f''' (x) = 10 \frac{d}{d x} ({(x - 3)}^{3 \cdot - 1})$

Step 3.1.2.2.2

Multiply $3$ by $- 1$ .

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

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 $g (x) = x - 3$ .

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Step 3.2.1

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

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

Step 3.2.2

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

$f''' (x) = 10 (- 3 u^{- 4} \frac{d}{d x} (x - 3))$

Step 3.2.3

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

$f''' (x) = 10 (- 3 {(x - 3)}^{- 4} \frac{d}{d x} (x - 3))$

Step 3.3

Differentiate.

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Step 3.3.1

Multiply $- 3$ by $10$ .

$f''' (x) = - 30 ({(x - 3)}^{- 4} \frac{d}{d x} (x - 3))$

Step 3.3.2

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

$f''' (x) = - 30 {(x - 3)}^{- 4} (\frac{d}{d x} (x) + \frac{d}{d x} (- 3))$

Step 3.3.3

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

$f''' (x) = - 30 {(x - 3)}^{- 4} (1 + \frac{d}{d x} (- 3))$

Step 3.3.4

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

$f''' (x) = - 30 {(x - 3)}^{- 4} (1 + 0)$

Step 3.3.5

Simplify the expression.

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Step 3.3.5.1

Add $1$ and $0$ .

$f''' (x) = - 30 {(x - 3)}^{- 4} \cdot 1$

Step 3.3.5.2

Multiply $- 30$ by $1$ .

$f''' (x) = - 30 {(x - 3)}^{- 4}$

Step 3.4

Simplify.

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Step 3.4.1

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

$f''' (x) = - 30 \frac{1}{{(x - 3)}^{4}}$

Step 3.4.2

Combine terms.

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Step 3.4.2.1

Combine $- 30$ and $\frac{1}{{(x - 3)}^{4}}$ .

$f''' (x) = \frac{- 30}{{(x - 3)}^{4}}$

Step 3.4.2.2

Move the negative in front of the fraction.

$f''' (x) = - \frac{30}{{(x - 3)}^{4}}$

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 $- 30$ is constant with respect to $x$ , the derivative of $- \frac{30}{{(x - 3)}^{4}}$ with respect to $x$ is $- 30 \frac{d}{d x} [\frac{1}{{(x - 3)}^{4}}]$ .

$f^{4} (x) = - 30 \frac{d}{d x} \frac{1}{{(x - 3)}^{4}}$

Step 4.1.2

Apply basic rules of exponents.

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Step 4.1.2.1

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

$f^{4} (x) = - 30 \frac{d}{d x} {({(x - 3)}^{4})}^{- 1}$

Step 4.1.2.2

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

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Step 4.1.2.2.1

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

$f^{4} (x) = - 30 \frac{d}{d x} {(x - 3)}^{4 \cdot - 1}$

Step 4.1.2.2.2

Multiply $4$ by $- 1$ .

$f^{4} (x) = - 30 \frac{d}{d x} {(x - 3)}^{- 4}$

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 $g (x) = x - 3$ .

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Step 4.2.1

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

$f^{4} (x) = - 30 (\frac{d}{d u} (u^{- 4}) \frac{d}{d x} (x - 3))$

Step 4.2.2

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

$f^{4} (x) = - 30 (- 4 u^{- 5} \frac{d}{d x} (x - 3))$

Step 4.2.3

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

$f^{4} (x) = - 30 (- 4 {(x - 3)}^{- 5} \frac{d}{d x} (x - 3))$

Step 4.3

Differentiate.

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Step 4.3.1

Multiply $- 4$ by $- 30$ .

$f^{4} (x) = 120 ({(x - 3)}^{- 5} \frac{d}{d x} (x - 3))$

Step 4.3.2

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

$f^{4} (x) = 120 {(x - 3)}^{- 5} (\frac{d}{d x} (x) + \frac{d}{d x} (- 3))$

Step 4.3.3

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

$f^{4} (x) = 120 {(x - 3)}^{- 5} (1 + \frac{d}{d x} (- 3))$

Step 4.3.4

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

$f^{4} (x) = 120 {(x - 3)}^{- 5} (1 + 0)$

Step 4.3.5

Simplify the expression.

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Step 4.3.5.1

Add $1$ and $0$ .

$f^{4} (x) = 120 {(x - 3)}^{- 5} \cdot 1$

Step 4.3.5.2

Multiply $120$ by $1$ .

$f^{4} (x) = 120 {(x - 3)}^{- 5}$

Step 4.4

Simplify.

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Step 4.4.1

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

$f^{4} (x) = 120 (\frac{1}{{(x - 3)}^{5}})$

Step 4.4.2

Combine $120$ and $\frac{1}{{(x - 3)}^{5}}$ .

$f^{4} (x) = \frac{120}{{(x - 3)}^{5}}$