Calculus Examples

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

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 $- 3 + \log_{\frac{1}{3}} (x)$ with respect to $x$ is $\frac{d}{d x} [- 3] + \frac{d}{d x} [\log_{\frac{1}{3}} (x)]$ .

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

Step 1.1.2

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

$0 + \frac{d}{d x} [\log_{\frac{1}{3}} (x)]$

Step 1.2

The derivative of $\log_{\frac{1}{3}} (x)$ with respect to $x$ is $\frac{1}{x \ln (\frac{1}{3})}$ .

$0 + \frac{1}{x \ln (\frac{1}{3})}$

Step 1.3

Add $0$ and $\frac{1}{x \ln (\frac{1}{3})}$ .

$\frac{1}{x \ln (\frac{1}{3})}$

Step 2

Find the second derivative of the function.

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Step 2.1

Since $\frac{1}{\ln (\frac{1}{3})}$ is constant with respect to $x$ , the derivative of $\frac{1}{x \ln (\frac{1}{3})}$ with respect to $x$ is $\frac{1}{\ln (\frac{1}{3})} \frac{d}{d x} [\frac{1}{x}]$ .

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

Step 2.2

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

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

Step 2.3

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

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

Step 2.4

Combine fractions.

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Step 2.4.1

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

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

Step 2.4.2

Simplify the expression.

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Step 2.4.2.1

Move $x^{- 2}$ to the denominator using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

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

Step 2.4.2.2

Reorder factors in $- \frac{1}{\ln (\frac{1}{3}) x^{2}}$ .

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

Step 3

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

$\frac{1}{x \ln (\frac{1}{3})} = 0$

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