Calculus Examples

$y = 4 x - \ln (x)$

Step 1

Find the first derivative.

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Step 1.1

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

$\frac{d}{d x} [4 x] + \frac{d}{d x} [- \ln (x)]$

Step 1.2

Evaluate $\frac{d}{d x} [4 x]$ .

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Step 1.2.1

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

$4 \frac{d}{d x} [x] + \frac{d}{d x} [- \ln (x)]$

Step 1.2.2

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

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

Step 1.2.3

Multiply $4$ by $1$ .

$4 + \frac{d}{d x} [- \ln (x)]$

Step 1.3

Evaluate $\frac{d}{d x} [- \ln (x)]$ .

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Step 1.3.1

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

$4 - \frac{d}{d x} [\ln (x)]$

Step 1.3.2

The derivative of $\ln (x)$ with respect to $x$ is $\frac{1}{x}$ .

$4 - \frac{1}{x}$

Step 1.4

Reorder terms.

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

Step 2

Find the second derivative.

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Step 2.1

By the Sum Rule, the derivative of $- \frac{1}{x} + 4$ with respect to $x$ is $\frac{d}{d x} [- \frac{1}{x}] + \frac{d}{d x} [4]$ .

$\frac{d}{d x} [- \frac{1}{x}] + \frac{d}{d x} [4]$

Step 2.2

Evaluate $\frac{d}{d x} [- \frac{1}{x}]$ .

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Step 2.2.1

Differentiate using the Product Rule which states that $\frac{d}{d x} [f (x) g (x)]$ is $f (x) \frac{d}{d x} [g (x)] + g (x) \frac{d}{d x} [f (x)]$ where $f (x) = - 1$ and $g (x) = \frac{1}{x}$ .

$- \frac{d}{d x} [\frac{1}{x}] + \frac{1}{x} \frac{d}{d x} [- 1] + \frac{d}{d x} [4]$

Step 2.2.2

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

$- \frac{d}{d x} [x^{- 1}] + \frac{1}{x} \frac{d}{d x} [- 1] + \frac{d}{d x} [4]$

Step 2.2.3

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

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

Step 2.2.4

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

$- - x^{- 2} + \frac{1}{x} \cdot 0 + \frac{d}{d x} [4]$

Step 2.2.5

Multiply $- 1$ by $- 1$ .

$1 x^{- 2} + \frac{1}{x} \cdot 0 + \frac{d}{d x} [4]$

Step 2.2.6

Multiply $x^{- 2}$ by $1$ .

$x^{- 2} + \frac{1}{x} \cdot 0 + \frac{d}{d x} [4]$

Step 2.2.7

Multiply $\frac{1}{x}$ by $0$ .

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

Step 2.2.8

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

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

Step 2.3

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

$x^{- 2} + 0$

Step 2.4

Simplify.

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Step 2.4.1

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

$\frac{1}{x^{2}} + 0$

Step 2.4.2

Add $\frac{1}{x^{2}}$ and $0$ .

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

Step 3

Find the third derivative.

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Step 3.1

Apply basic rules of exponents.

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Step 3.1.1

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

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

Step 3.1.2

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

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Step 3.1.2.1

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

$\frac{d}{d x} [x^{2 \cdot - 1}]$

Step 3.1.2.2

Multiply $2$ by $- 1$ .

$\frac{d}{d x} [x^{- 2}]$

Step 3.2

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

$- 2 x^{- 3}$

Step 3.3

Simplify.

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Step 3.3.1

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

$- 2 \frac{1}{x^{3}}$

Step 3.3.2

Combine terms.

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Step 3.3.2.1

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

$\frac{- 2}{x^{3}}$

Step 3.3.2.2

Move the negative in front of the fraction.

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

Step 4

Find the fourth derivative.

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Step 4.1

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

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

Step 4.2

Apply basic rules of exponents.

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Step 4.2.1

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

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

Step 4.2.2

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

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Step 4.2.2.1

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

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

Step 4.2.2.2

Multiply $3$ by $- 1$ .

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

Step 4.3

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

$- 2 (- 3 x^{- 4})$

Step 4.4

Multiply $- 3$ by $- 2$ .

$6 x^{- 4}$

Step 4.5

Simplify.

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Step 4.5.1

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

$6 \frac{1}{x^{4}}$

Step 4.5.2

Combine $6$ and $\frac{1}{x^{4}}$ .

$f^{4} (x) = \frac{6}{x^{4}}$