Calculus Examples

$y = 4 x - 6 \ln (x)$

Step 1

Write $y = 4 x - 6 \ln (x)$ as a function.

$f (x) = 4 x - 6 \ln (x)$

Step 2

Find the first derivative of the function.

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Step 2.1

By the Sum Rule, the derivative of $4 x - 6 \ln (x)$ with respect to $x$ is $\frac{d}{d x} [4 x] + \frac{d}{d x} [- 6 \ln (x)]$ .

$\frac{d}{d x} [4 x] + \frac{d}{d x} [- 6 \ln (x)]$

Step 2.2

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

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Step 2.2.1

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

$4 \frac{d}{d x} [x] + \frac{d}{d x} [- 6 \ln (x)]$

Step 2.2.2

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

$4 \cdot 1 + \frac{d}{d x} [- 6 \ln (x)]$

Step 2.2.3

Multiply $4$ by $1$ .

$4 + \frac{d}{d x} [- 6 \ln (x)]$

Step 2.3

Evaluate $\frac{d}{d x} [- 6 \ln (x)]$ .

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Step 2.3.1

Since $- 6$ is constant with respect to $x$ , the derivative of $- 6 \ln (x)$ with respect to $x$ is $- 6 \frac{d}{d x} [\ln (x)]$ .

$4 - 6 \frac{d}{d x} [\ln (x)]$

Step 2.3.2

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

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

Step 2.3.3

Combine $- 6$ and $\frac{1}{x}$ .

$4 + \frac{- 6}{x}$

Step 2.3.4

Move the negative in front of the fraction.

$4 - \frac{6}{x}$

Step 2.4

Reorder terms.

$- \frac{6}{x} + 4$

Step 3

Find the second derivative of the function.

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Step 3.1

By the Sum Rule, the derivative of $- \frac{6}{x} + 4$ with respect to $x$ is $\frac{d}{d x} [- \frac{6}{x}] + \frac{d}{d x} [4]$ .

$f'' (x) = \frac{d}{d x} (- \frac{6}{x}) + \frac{d}{d x} (4)$

Step 3.2

Evaluate $\frac{d}{d x} [- \frac{6}{x}]$ .

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Step 3.2.1

Since $- 6$ is constant with respect to $x$ , the derivative of $- \frac{6}{x}$ with respect to $x$ is $- 6 \frac{d}{d x} [\frac{1}{x}]$ .

$f'' (x) = - 6 \frac{d}{d x} \frac{1}{x} + \frac{d}{d x} (4)$

Step 3.2.2

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

$f'' (x) = - 6 \frac{d}{d x} x^{- 1} + \frac{d}{d x} (4)$

Step 3.2.3

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

$f'' (x) = - 6 (- x^{- 2}) + \frac{d}{d x} (4)$

Step 3.2.4

Multiply $- 1$ by $- 6$ .

$f'' (x) = 6 x^{- 2} + \frac{d}{d x} (4)$

Step 3.3

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

$f'' (x) = 6 x^{- 2} + 0$

Step 3.4

Simplify.

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Step 3.4.1

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

$f'' (x) = 6 (\frac{1}{x^{2}}) + 0$

Step 3.4.2

Combine terms.

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Step 3.4.2.1

Combine $6$ and $\frac{1}{x^{2}}$ .

$f'' (x) = \frac{6}{x^{2}} + 0$

Step 3.4.2.2

Add $\frac{6}{x^{2}}$ and $0$ .

$f'' (x) = \frac{6}{x^{2}}$

Step 4

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

$- \frac{6}{x} + 4 = 0$

Step 5

Find the first derivative.

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Step 5.1

Find the first derivative.

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Step 5.1.1

By the Sum Rule, the derivative of $4 x - 6 \ln (x)$ with respect to $x$ is $\frac{d}{d x} [4 x] + \frac{d}{d x} [- 6 \ln (x)]$ .

$\frac{d}{d x} [4 x] + \frac{d}{d x} [- 6 \ln (x)]$

Step 5.1.2

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

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Step 5.1.2.1

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

$4 \frac{d}{d x} [x] + \frac{d}{d x} [- 6 \ln (x)]$

Step 5.1.2.2

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

$4 \cdot 1 + \frac{d}{d x} [- 6 \ln (x)]$

Step 5.1.2.3

Multiply $4$ by $1$ .

$4 + \frac{d}{d x} [- 6 \ln (x)]$

Step 5.1.3

Evaluate $\frac{d}{d x} [- 6 \ln (x)]$ .

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Step 5.1.3.1

Since $- 6$ is constant with respect to $x$ , the derivative of $- 6 \ln (x)$ with respect to $x$ is $- 6 \frac{d}{d x} [\ln (x)]$ .

$4 - 6 \frac{d}{d x} [\ln (x)]$

Step 5.1.3.2

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

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

Step 5.1.3.3

Combine $- 6$ and $\frac{1}{x}$ .

$4 + \frac{- 6}{x}$

Step 5.1.3.4

Move the negative in front of the fraction.

$4 - \frac{6}{x}$

Step 5.1.4

Reorder terms.

$f' (x) = - \frac{6}{x} + 4$

Step 5.2

The first derivative of $f (x)$ with respect to $x$ is $- \frac{6}{x} + 4$ .

$- \frac{6}{x} + 4$

Step 6

Set the first derivative equal to $0$ then solve the equation $- \frac{6}{x} + 4 = 0$ .

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Step 6.1

Set the first derivative equal to $0$ .

$- \frac{6}{x} + 4 = 0$

Step 6.2

Subtract $4$ from both sides of the equation.

$- \frac{6}{x} = - 4$

Step 6.3

Find the LCD of the terms in the equation.

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Step 6.3.1

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

$x, 1$

Step 6.3.2

The LCM of one and any expression is the expression.

$x$

Step 6.4

Multiply each term in $- \frac{6}{x} = - 4$ by $x$ to eliminate the fractions.

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Step 6.4.1

Multiply each term in $- \frac{6}{x} = - 4$ by $x$ .

$- \frac{6}{x} x = - 4 x$

Step 6.4.2

Simplify the left side.

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Step 6.4.2.1

Cancel the common factor of $x$ .

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Step 6.4.2.1.1

Move the leading negative in $- \frac{6}{x}$ into the numerator.

$\frac{- 6}{x} x = - 4 x$

Step 6.4.2.1.2

Cancel the common factor.

$\frac{- 6}{x} x = - 4 x$

Step 6.4.2.1.3

Rewrite the expression.

$- 6 = - 4 x$

Step 6.5

Solve the equation.

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Step 6.5.1

Rewrite the equation as $- 4 x = - 6$ .

$- 4 x = - 6$

Step 6.5.2

Divide each term in $- 4 x = - 6$ by $- 4$ and simplify.

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Step 6.5.2.1

Divide each term in $- 4 x = - 6$ by $- 4$ .

$\frac{- 4 x}{- 4} = \frac{- 6}{- 4}$

Step 6.5.2.2

Simplify the left side.

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Step 6.5.2.2.1

Cancel the common factor of $- 4$ .

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Step 6.5.2.2.1.1

Cancel the common factor.

$\frac{- 4 x}{- 4} = \frac{- 6}{- 4}$

Step 6.5.2.2.1.2

Divide $x$ by $1$ .

$x = \frac{- 6}{- 4}$

Step 6.5.2.3

Simplify the right side.

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Step 6.5.2.3.1

Cancel the common factor of $- 6$ and $- 4$ .

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Step 6.5.2.3.1.1

Factor $- 2$ out of $- 6$ .

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

Step 6.5.2.3.1.2

Cancel the common factors.

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Step 6.5.2.3.1.2.1

Factor $- 2$ out of $- 4$ .

$x = \frac{- 2 \cdot 3}{- 2 \cdot 2}$

Step 6.5.2.3.1.2.2

Cancel the common factor.

$x = \frac{- 2 \cdot 3}{- 2 \cdot 2}$

Step 6.5.2.3.1.2.3

Rewrite the expression.

$x = \frac{3}{2}$

Step 7

Find the values where the derivative is undefined.

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Step 7.1

Set the denominator in $\frac{6}{x}$ equal to $0$ to find where the expression is undefined.

$x = 0$

Step 8

Critical points to evaluate.

$x = \frac{3}{2}$

Step 9

Evaluate the second derivative at $x = \frac{3}{2}$ . If the second derivative is positive, then this is a local minimum. If it is negative, then this is a local maximum.

$\frac{6}{{(\frac{3}{2})}^{2}}$

Step 10

Evaluate the second derivative.

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Step 10.1

Simplify the denominator.

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Step 10.1.1

Apply the product rule to $\frac{3}{2}$ .

$\frac{6}{\frac{3^{2}}{2^{2}}}$

Step 10.1.2

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

$\frac{6}{\frac{9}{2^{2}}}$

Step 10.1.3

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

$\frac{6}{\frac{9}{4}}$

Step 10.2

Multiply the numerator by the reciprocal of the denominator.

$6 (\frac{4}{9})$

Step 10.3

Cancel the common factor of $3$ .

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Step 10.3.1

Factor $3$ out of $6$ .

$3 (2) \frac{4}{9}$

Step 10.3.2

Factor $3$ out of $9$ .

$3 \cdot 2 \frac{4}{3 \cdot 3}$

Step 10.3.3

Cancel the common factor.

$3 \cdot 2 \frac{4}{3 \cdot 3}$

Step 10.3.4

Rewrite the expression.

$2 (\frac{4}{3})$

Step 10.4

Combine $2$ and $\frac{4}{3}$ .

$\frac{2 \cdot 4}{3}$

Step 10.5

Multiply $2$ by $4$ .

$\frac{8}{3}$

Step 11

$x = \frac{3}{2}$ is a local minimum because the value of the second derivative is positive. This is referred to as the second derivative test.

$x = \frac{3}{2}$ is a local minimum

Step 12

Find the y-value when $x = \frac{3}{2}$ .

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Step 12.1

Replace the variable $x$ with $\frac{3}{2}$ in the expression.

$f (\frac{3}{2}) = 4 (\frac{3}{2}) - 6 \ln (\frac{3}{2})$

Step 12.2

Simplify the result.

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Step 12.2.1

Simplify each term.

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Step 12.2.1.1

Cancel the common factor of $2$ .

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Step 12.2.1.1.1

Factor $2$ out of $4$ .

$f (\frac{3}{2}) = 2 (2) (\frac{3}{2}) - 6 \ln (\frac{3}{2})$

Step 12.2.1.1.2

Cancel the common factor.

$f (\frac{3}{2}) = 2 \cdot (2 (\frac{3}{2})) - 6 \ln (\frac{3}{2})$

Step 12.2.1.1.3

Rewrite the expression.

$f (\frac{3}{2}) = 2 \cdot 3 - 6 \ln (\frac{3}{2})$

Step 12.2.1.2

Multiply $2$ by $3$ .

$f (\frac{3}{2}) = 6 - 6 \ln (\frac{3}{2})$

Step 12.2.1.3

Simplify $- 6 \ln (\frac{3}{2})$ by moving $6$ inside the logarithm.

$f (\frac{3}{2}) = 6 - \ln ({(\frac{3}{2})}^{6})$

Step 12.2.1.4

Apply the product rule to $\frac{3}{2}$ .

$f (\frac{3}{2}) = 6 - \ln (\frac{3^{6}}{2^{6}})$

Step 12.2.1.5

Raise $3$ to the power of $6$ .

$f (\frac{3}{2}) = 6 - \ln (\frac{729}{2^{6}})$

Step 12.2.1.6

Raise $2$ to the power of $6$ .

$f (\frac{3}{2}) = 6 - \ln (\frac{729}{64})$

Step 12.2.2

The final answer is $6 - \ln (\frac{729}{64})$ .

$y = 6 - \ln (\frac{729}{64})$

Step 13

These are the local extrema for $f (x) = 4 x - 6 \ln (x)$ .

$(\frac{3}{2}, 6 - \ln (\frac{729}{64}))$ is a local minima

Step 14