Calculus Examples

$f (x) = 3 x - \ln (x)$

Step 1

Find the first derivative of the function.

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Step 1.1

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

$\frac{d}{d x} [3 x] + \frac{d}{d x} [- \ln (x)]$

Step 1.2

Evaluate $\frac{d}{d x} [3 x]$ .

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Step 1.2.1

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

$3 \frac{d}{d x} [x] + \frac{d}{d x} [- \ln (x)]$

Step 1.2.2

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

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

Step 1.2.3

Multiply $3$ by $1$ .

$3 + \frac{d}{d x} [- \ln (x)]$

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

$3 - \frac{d}{d x} [\ln (x)]$

Step 1.3.2

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

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

Step 1.4

Reorder terms.

$- \frac{1}{x} + 3$

Step 2

Find the second derivative of the function.

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Step 2.1

By the Sum Rule, the derivative of $- \frac{1}{x} + 3$ with respect to $x$ is $\frac{d}{d x} [- \frac{1}{x}] + \frac{d}{d x} [3]$ .

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

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 $g (x) = \frac{1}{x}$ .

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

Step 2.2.2

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

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

Step 2.2.3

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

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

Step 2.2.4

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

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

Step 2.2.5

Multiply $- 1$ by $- 1$ .

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

Step 2.2.6

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

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

Step 2.2.7

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

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

Step 2.2.8

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

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

Step 2.3

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

$f'' (x) = x^{- 2} + 0$

Step 2.4

Simplify.

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Step 2.4.1

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

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

Step 2.4.2

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

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

Step 3

To find the local maximum and minimum values of the function, set the derivative equal to $0$ and solve.

$- \frac{1}{x} + 3 = 0$

Step 4

Find the first derivative.

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Step 4.1

Find the first derivative.

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Step 4.1.1

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

$\frac{d}{d x} [3 x] + \frac{d}{d x} [- \ln (x)]$

Step 4.1.2

Evaluate $\frac{d}{d x} [3 x]$ .

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Step 4.1.2.1

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

$3 \frac{d}{d x} [x] + \frac{d}{d x} [- \ln (x)]$

Step 4.1.2.2

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

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

Step 4.1.2.3

Multiply $3$ by $1$ .

$3 + \frac{d}{d x} [- \ln (x)]$

Step 4.1.3

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

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Step 4.1.3.1

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

$3 - \frac{d}{d x} [\ln (x)]$

Step 4.1.3.2

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

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

Step 4.1.4

Reorder terms.

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

Step 4.2

The first derivative of $f (x)$ with respect to $x$ is $- \frac{1}{x} + 3$ .

$- \frac{1}{x} + 3$

Step 5

Set the first derivative equal to $0$ then solve the equation $- \frac{1}{x} + 3 = 0$ .

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Step 5.1

Set the first derivative equal to $0$ .

$- \frac{1}{x} + 3 = 0$

Step 5.2

Subtract $3$ from both sides of the equation.

$- \frac{1}{x} = - 3$

Step 5.3

Find the LCD of the terms in the equation.

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Step 5.3.1

Finding the LCD of a list of values is the same as finding the LCM of the denominators of those values.

$x, 1$

Step 5.3.2

The LCM of one and any expression is the expression.

$x$

Step 5.4

Multiply each term in $- \frac{1}{x} = - 3$ by $x$ to eliminate the fractions.

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Step 5.4.1

Multiply each term in $- \frac{1}{x} = - 3$ by $x$ .

$- \frac{1}{x} x = - 3 x$

Step 5.4.2

Simplify the left side.

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Step 5.4.2.1

Cancel the common factor of $x$ .

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Step 5.4.2.1.1

Move the leading negative in $- \frac{1}{x}$ into the numerator.

$\frac{- 1}{x} x = - 3 x$

Step 5.4.2.1.2

Cancel the common factor.

$\frac{- 1}{x} x = - 3 x$

Step 5.4.2.1.3

Rewrite the expression.

$- 1 = - 3 x$

Step 5.5

Solve the equation.

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Step 5.5.1

Rewrite the equation as $- 3 x = - 1$ .

$- 3 x = - 1$

Step 5.5.2

Divide each term in $- 3 x = - 1$ by $- 3$ and simplify.

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Step 5.5.2.1

Divide each term in $- 3 x = - 1$ by $- 3$ .

$\frac{- 3 x}{- 3} = \frac{- 1}{- 3}$

Step 5.5.2.2

Simplify the left side.

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Step 5.5.2.2.1

Cancel the common factor of $- 3$ .

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Step 5.5.2.2.1.1

Cancel the common factor.

$\frac{- 3 x}{- 3} = \frac{- 1}{- 3}$

Step 5.5.2.2.1.2

Divide $x$ by $1$ .

$x = \frac{- 1}{- 3}$

Step 5.5.2.3

Simplify the right side.

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Step 5.5.2.3.1

Dividing two negative values results in a positive value.

$x = \frac{1}{3}$

Step 6

Find the values where the derivative is undefined.

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Step 6.1

Set the denominator in $\frac{1}{x}$ equal to $0$ to find where the expression is undefined.

$x = 0$

Step 7

Critical points to evaluate.

$x = \frac{1}{3}$

Step 8

Evaluate the second derivative at $x = \frac{1}{3}$ . If the second derivative is positive, then this is a local minimum. If it is negative, then this is a local maximum.

$\frac{1}{{(\frac{1}{3})}^{2}}$

Step 9

Evaluate the second derivative.

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Step 9.1

Simplify the denominator.

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Step 9.1.1

Apply the product rule to $\frac{1}{3}$ .

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

Step 9.1.2

One to any power is one.

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

Step 9.1.3

Raise $3$ to the power of $2$ .

$\frac{1}{\frac{1}{9}}$

Step 9.2

Multiply the numerator by the reciprocal of the denominator.

$1 \cdot 9$

Step 9.3

Multiply $9$ by $1$ .

$9$

Step 10

$x = \frac{1}{3}$ is a local minimum because the value of the second derivative is positive. This is referred to as the second derivative test.

$x = \frac{1}{3}$ is a local minimum

Step 11

Find the y-value when $x = \frac{1}{3}$ .

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Step 11.1

Replace the variable $x$ with $\frac{1}{3}$ in the expression.

$f (\frac{1}{3}) = 3 (\frac{1}{3}) - \ln (\frac{1}{3})$

Step 11.2

Simplify the result.

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Step 11.2.1

Cancel the common factor of $3$ .

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Step 11.2.1.1

Cancel the common factor.

$f (\frac{1}{3}) = 3 (\frac{1}{3}) - \ln (\frac{1}{3})$

Step 11.2.1.2

Rewrite the expression.

$f (\frac{1}{3}) = 1 - \ln (\frac{1}{3})$

Step 11.2.2

The final answer is $1 - \ln (\frac{1}{3})$ .

$y = 1 - \ln (\frac{1}{3})$

Step 12

These are the local extrema for $f (x) = 3 x - \ln (x)$ .

$(\frac{1}{3}, 1 - \ln (\frac{1}{3}))$ is a local minima

Step 13