Calculus Examples

$f (x) = \frac{x}{4 - 7 x}$

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 $f (x) = x$ and $g (x) = 4 - 7 x$ .

$\frac{(4 - 7 x) \frac{d}{d x} [x] - x \frac{d}{d x} [4 - 7 x]}{{(4 - 7 x)}^{2}}$

Step 1.2

Differentiate.

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Step 1.2.1

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

$\frac{(4 - 7 x) \cdot 1 - x \frac{d}{d x} [4 - 7 x]}{{(4 - 7 x)}^{2}}$

Step 1.2.2

Multiply $4 - 7 x$ by $1$ .

$\frac{4 - 7 x - x \frac{d}{d x} [4 - 7 x]}{{(4 - 7 x)}^{2}}$

Step 1.2.3

By the Sum Rule, the derivative of $4 - 7 x$ with respect to $x$ is $\frac{d}{d x} [4] + \frac{d}{d x} [- 7 x]$ .

$\frac{4 - 7 x - x (\frac{d}{d x} [4] + \frac{d}{d x} [- 7 x])}{{(4 - 7 x)}^{2}}$

Step 1.2.4

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

$\frac{4 - 7 x - x (0 + \frac{d}{d x} [- 7 x])}{{(4 - 7 x)}^{2}}$

Step 1.2.5

Add $0$ and $\frac{d}{d x} [- 7 x]$ .

$\frac{4 - 7 x - x \frac{d}{d x} [- 7 x]}{{(4 - 7 x)}^{2}}$

Step 1.2.6

Since $- 7$ is constant with respect to $x$ , the derivative of $- 7 x$ with respect to $x$ is $- 7 \frac{d}{d x} [x]$ .

$\frac{4 - 7 x - x (- 7 \frac{d}{d x} [x])}{{(4 - 7 x)}^{2}}$

Step 1.2.7

Multiply $- 7$ by $- 1$ .

$\frac{4 - 7 x + 7 x \frac{d}{d x} [x]}{{(4 - 7 x)}^{2}}$

Step 1.2.8

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

$\frac{4 - 7 x + 7 x \cdot 1}{{(4 - 7 x)}^{2}}$

Step 1.2.9

Simplify by adding terms.

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Step 1.2.9.1

Multiply $7$ by $1$ .

$\frac{4 - 7 x + 7 x}{{(4 - 7 x)}^{2}}$

Step 1.2.9.2

Add $- 7 x$ and $7 x$ .

$\frac{4 + 0}{{(4 - 7 x)}^{2}}$

Step 1.2.9.3

Simplify the expression.

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Step 1.2.9.3.1

Add $4$ and $0$ .

$\frac{4}{{(4 - 7 x)}^{2}}$

Step 1.2.9.3.2

Reorder terms.

$f' (x) = \frac{4}{{(- 7 x + 4)}^{2}}$

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 $4$ is constant with respect to $x$ , the derivative of $\frac{4}{{(- 7 x + 4)}^{2}}$ with respect to $x$ is $4 \frac{d}{d x} [\frac{1}{{(- 7 x + 4)}^{2}}]$ .

$4 \frac{d}{d x} [\frac{1}{{(- 7 x + 4)}^{2}}]$

Step 2.1.2

Apply basic rules of exponents.

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Step 2.1.2.1

Rewrite $\frac{1}{{(- 7 x + 4)}^{2}}$ as ${({(- 7 x + 4)}^{2})}^{- 1}$ .

$4 \frac{d}{d x} [{({(- 7 x + 4)}^{2})}^{- 1}]$

Step 2.1.2.2

Multiply the exponents in ${({(- 7 x + 4)}^{2})}^{- 1}$ .

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Step 2.1.2.2.1

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

$4 \frac{d}{d x} [{(- 7 x + 4)}^{2 \cdot - 1}]$

Step 2.1.2.2.2

Multiply $2$ by $- 1$ .

$4 \frac{d}{d x} [{(- 7 x + 4)}^{- 2}]$

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 $g (x) = - 7 x + 4$ .

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Step 2.2.1

To apply the Chain Rule, set $u$ as $- 7 x + 4$ .

$4 (\frac{d}{d u} [u^{- 2}] \frac{d}{d x} [- 7 x + 4])$

Step 2.2.2

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

$4 (- 2 u^{- 3} \frac{d}{d x} [- 7 x + 4])$

Step 2.2.3

Replace all occurrences of $u$ with $- 7 x + 4$ .

$4 (- 2 {(- 7 x + 4)}^{- 3} \frac{d}{d x} [- 7 x + 4])$

Step 2.3

Differentiate.

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Step 2.3.1

Multiply $- 2$ by $4$ .

$- 8 ({(- 7 x + 4)}^{- 3} \frac{d}{d x} [- 7 x + 4])$

Step 2.3.2

By the Sum Rule, the derivative of $- 7 x + 4$ with respect to $x$ is $\frac{d}{d x} [- 7 x] + \frac{d}{d x} [4]$ .

$- 8 {(- 7 x + 4)}^{- 3} (\frac{d}{d x} [- 7 x] + \frac{d}{d x} [4])$

Step 2.3.3

Since $- 7$ is constant with respect to $x$ , the derivative of $- 7 x$ with respect to $x$ is $- 7 \frac{d}{d x} [x]$ .

$- 8 {(- 7 x + 4)}^{- 3} (- 7 \frac{d}{d x} [x] + \frac{d}{d x} [4])$

Step 2.3.4

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

$- 8 {(- 7 x + 4)}^{- 3} (- 7 \cdot 1 + \frac{d}{d x} [4])$

Step 2.3.5

Multiply $- 7$ by $1$ .

$- 8 {(- 7 x + 4)}^{- 3} (- 7 + \frac{d}{d x} [4])$

Step 2.3.6

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

$- 8 {(- 7 x + 4)}^{- 3} (- 7 + 0)$

Step 2.3.7

Simplify the expression.

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Step 2.3.7.1

Add $- 7$ and $0$ .

$- 8 {(- 7 x + 4)}^{- 3} \cdot - 7$

Step 2.3.7.2

Multiply $- 7$ by $- 8$ .

$56 {(- 7 x + 4)}^{- 3}$

Step 2.4

Simplify.

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Step 2.4.1

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

$56 \frac{1}{{(- 7 x + 4)}^{3}}$

Step 2.4.2

Combine $56$ and $\frac{1}{{(- 7 x + 4)}^{3}}$ .

$f'' (x) = \frac{56}{{(- 7 x + 4)}^{3}}$

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 $56$ is constant with respect to $x$ , the derivative of $\frac{56}{{(- 7 x + 4)}^{3}}$ with respect to $x$ is $56 \frac{d}{d x} [\frac{1}{{(- 7 x + 4)}^{3}}]$ .

$56 \frac{d}{d x} [\frac{1}{{(- 7 x + 4)}^{3}}]$

Step 3.1.2

Apply basic rules of exponents.

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Step 3.1.2.1

Rewrite $\frac{1}{{(- 7 x + 4)}^{3}}$ as ${({(- 7 x + 4)}^{3})}^{- 1}$ .

$56 \frac{d}{d x} [{({(- 7 x + 4)}^{3})}^{- 1}]$

Step 3.1.2.2

Multiply the exponents in ${({(- 7 x + 4)}^{3})}^{- 1}$ .

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Step 3.1.2.2.1

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

$56 \frac{d}{d x} [{(- 7 x + 4)}^{3 \cdot - 1}]$

Step 3.1.2.2.2

Multiply $3$ by $- 1$ .

$56 \frac{d}{d x} [{(- 7 x + 4)}^{- 3}]$

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 $g (x) = - 7 x + 4$ .

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Step 3.2.1

To apply the Chain Rule, set $u$ as $- 7 x + 4$ .

$56 (\frac{d}{d u} [u^{- 3}] \frac{d}{d x} [- 7 x + 4])$

Step 3.2.2

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

$56 (- 3 u^{- 4} \frac{d}{d x} [- 7 x + 4])$

Step 3.2.3

Replace all occurrences of $u$ with $- 7 x + 4$ .

$56 (- 3 {(- 7 x + 4)}^{- 4} \frac{d}{d x} [- 7 x + 4])$

Step 3.3

Differentiate.

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Step 3.3.1

Multiply $- 3$ by $56$ .

$- 168 ({(- 7 x + 4)}^{- 4} \frac{d}{d x} [- 7 x + 4])$

Step 3.3.2

By the Sum Rule, the derivative of $- 7 x + 4$ with respect to $x$ is $\frac{d}{d x} [- 7 x] + \frac{d}{d x} [4]$ .

$- 168 {(- 7 x + 4)}^{- 4} (\frac{d}{d x} [- 7 x] + \frac{d}{d x} [4])$

Step 3.3.3

Since $- 7$ is constant with respect to $x$ , the derivative of $- 7 x$ with respect to $x$ is $- 7 \frac{d}{d x} [x]$ .

$- 168 {(- 7 x + 4)}^{- 4} (- 7 \frac{d}{d x} [x] + \frac{d}{d x} [4])$

Step 3.3.4

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

$- 168 {(- 7 x + 4)}^{- 4} (- 7 \cdot 1 + \frac{d}{d x} [4])$

Step 3.3.5

Multiply $- 7$ by $1$ .

$- 168 {(- 7 x + 4)}^{- 4} (- 7 + \frac{d}{d x} [4])$

Step 3.3.6

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

$- 168 {(- 7 x + 4)}^{- 4} (- 7 + 0)$

Step 3.3.7

Simplify the expression.

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Step 3.3.7.1

Add $- 7$ and $0$ .

$- 168 {(- 7 x + 4)}^{- 4} \cdot - 7$

Step 3.3.7.2

Multiply $- 7$ by $- 168$ .

$1176 {(- 7 x + 4)}^{- 4}$

Step 3.4

Simplify.

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Step 3.4.1

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

$1176 \frac{1}{{(- 7 x + 4)}^{4}}$

Step 3.4.2

Combine $1176$ and $\frac{1}{{(- 7 x + 4)}^{4}}$ .

$f''' (x) = \frac{1176}{{(- 7 x + 4)}^{4}}$

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 $1176$ is constant with respect to $x$ , the derivative of $\frac{1176}{{(- 7 x + 4)}^{4}}$ with respect to $x$ is $1176 \frac{d}{d x} [\frac{1}{{(- 7 x + 4)}^{4}}]$ .

$1176 \frac{d}{d x} [\frac{1}{{(- 7 x + 4)}^{4}}]$

Step 4.1.2

Apply basic rules of exponents.

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Step 4.1.2.1

Rewrite $\frac{1}{{(- 7 x + 4)}^{4}}$ as ${({(- 7 x + 4)}^{4})}^{- 1}$ .

$1176 \frac{d}{d x} [{({(- 7 x + 4)}^{4})}^{- 1}]$

Step 4.1.2.2

Multiply the exponents in ${({(- 7 x + 4)}^{4})}^{- 1}$ .

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Step 4.1.2.2.1

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

$1176 \frac{d}{d x} [{(- 7 x + 4)}^{4 \cdot - 1}]$

Step 4.1.2.2.2

Multiply $4$ by $- 1$ .

$1176 \frac{d}{d x} [{(- 7 x + 4)}^{- 4}]$

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 $g (x) = - 7 x + 4$ .

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Step 4.2.1

To apply the Chain Rule, set $u$ as $- 7 x + 4$ .

$1176 (\frac{d}{d u} [u^{- 4}] \frac{d}{d x} [- 7 x + 4])$

Step 4.2.2

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

$1176 (- 4 u^{- 5} \frac{d}{d x} [- 7 x + 4])$

Step 4.2.3

Replace all occurrences of $u$ with $- 7 x + 4$ .

$1176 (- 4 {(- 7 x + 4)}^{- 5} \frac{d}{d x} [- 7 x + 4])$

Step 4.3

Differentiate.

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Step 4.3.1

Multiply $- 4$ by $1176$ .

$- 4704 ({(- 7 x + 4)}^{- 5} \frac{d}{d x} [- 7 x + 4])$

Step 4.3.2

By the Sum Rule, the derivative of $- 7 x + 4$ with respect to $x$ is $\frac{d}{d x} [- 7 x] + \frac{d}{d x} [4]$ .

$- 4704 {(- 7 x + 4)}^{- 5} (\frac{d}{d x} [- 7 x] + \frac{d}{d x} [4])$

Step 4.3.3

Since $- 7$ is constant with respect to $x$ , the derivative of $- 7 x$ with respect to $x$ is $- 7 \frac{d}{d x} [x]$ .

$- 4704 {(- 7 x + 4)}^{- 5} (- 7 \frac{d}{d x} [x] + \frac{d}{d x} [4])$

Step 4.3.4

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

$- 4704 {(- 7 x + 4)}^{- 5} (- 7 \cdot 1 + \frac{d}{d x} [4])$

Step 4.3.5

Multiply $- 7$ by $1$ .

$- 4704 {(- 7 x + 4)}^{- 5} (- 7 + \frac{d}{d x} [4])$

Step 4.3.6

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

$- 4704 {(- 7 x + 4)}^{- 5} (- 7 + 0)$

Step 4.3.7

Simplify the expression.

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Step 4.3.7.1

Add $- 7$ and $0$ .

$- 4704 {(- 7 x + 4)}^{- 5} \cdot - 7$

Step 4.3.7.2

Multiply $- 7$ by $- 4704$ .

$32928 {(- 7 x + 4)}^{- 5}$

Step 4.4

Simplify.

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Step 4.4.1

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

$32928 \frac{1}{{(- 7 x + 4)}^{5}}$

Step 4.4.2

Combine $32928$ and $\frac{1}{{(- 7 x + 4)}^{5}}$ .

$f^{4} (x) = \frac{32928}{{(- 7 x + 4)}^{5}}$

Step 5

The fourth derivative of $f (x)$ with respect to $x$ is $\frac{32928}{{(- 7 x + 4)}^{5}}$ .

$\frac{32928}{{(- 7 x + 4)}^{5}}$