Calculus Examples

$f (x) = 4 x + \ln (x)$

Step 1

Find the first derivative of the function.

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Step 1.1

By the Sum Rule, the derivative of $4 x + \ln (x)$ 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

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

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

Step 1.4

Reorder terms.

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

Step 2

Find the second derivative of the function.

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Step 2.1

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

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

Step 2.2

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

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Step 2.2.1

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

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

Step 2.2.2

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

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

Step 2.3

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

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

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.

$\frac{1}{x} + 4 = 0$

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