Calculus Examples

$f (x) = 1 - 6 \ln (x)$

Step 1

Find the first derivative of the function.

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Step 1.1

Differentiate.

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Step 1.1.1

By the Sum Rule, the derivative of $1 - 6 \ln (x)$ with respect to $x$ is $\frac{d}{d x} [1] + \frac{d}{d x} [- 6 \ln (x)]$ .

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

Step 1.1.2

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

$0 + \frac{d}{d x} [- 6 \ln (x)]$

Step 1.2

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

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Step 1.2.1

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

$0 - 6 \frac{d}{d x} [\ln (x)]$

Step 1.2.2

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

$0 - 6 \frac{1}{x}$

Step 1.2.3

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

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

Step 1.2.4

Move the negative in front of the fraction.

$0 - \frac{6}{x}$

Step 1.3

Subtract $\frac{6}{x}$ from $0$ .

$- \frac{6}{x}$

Step 2

Find the second derivative of the function.

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Step 2.1

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

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

Step 2.2

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

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

Step 2.3

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

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

Step 2.4

Multiply $- 1$ by $- 6$ .

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

Step 2.5

Simplify.

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Step 2.5.1

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

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

Step 2.5.2

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

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

Step 3

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

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

Step 4

Since there is no value of $x$ that makes the first derivative equal to $0$ , there are no local extrema.

No Local Extrema

Step 5

No Local Extrema

Step 6