Calculus Examples

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

Step 1

Find the $x$ values where the second derivative is equal to $0$ .

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Step 1.1

Find the second derivative.

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Step 1.1.1

Find the first derivative.

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Step 1.1.1.1

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 $f (x) = x$ and $g (x) = \ln (x)$ .

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

Step 1.1.1.2

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

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

Step 1.1.1.3

Differentiate using the Power Rule.

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Step 1.1.1.3.1

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

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

Step 1.1.1.3.2

Cancel the common factor of $x$ .

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Step 1.1.1.3.2.1

Cancel the common factor.

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

Step 1.1.1.3.2.2

Rewrite the expression.

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

Step 1.1.1.3.3

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

$1 + \ln (x) \cdot 1$

Step 1.1.1.3.4

Multiply $\ln (x)$ by $1$ .

$f' (x) = 1 + \ln (x)$

Step 1.1.2

Find the second derivative.

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Step 1.1.2.1

Differentiate.

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Step 1.1.2.1.1

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

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

Step 1.1.2.1.2

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

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

Step 1.1.2.2

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

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

Step 1.1.2.3

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

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

Step 1.1.3

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

$\frac{1}{x}$

Step 1.2

Set the second derivative equal to $0$ then solve the equation $\frac{1}{x} = 0$ .

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Step 1.2.1

Set the second derivative equal to $0$ .

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

Step 1.2.2

Set the numerator equal to zero.

$1 = 0$

Step 1.2.3

Since $1 \neq 0$ , there are no solutions.

No solution

Step 2

Find the domain of $f (x) = x \ln (x)$ .

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Step 2.1

Set the argument in $\ln (x)$ greater than $0$ to find where the expression is defined.

$x > 0$

Step 2.2

The domain is all values of $x$ that make the expression defined.

Interval Notation:

$(0, \infty)$

Set-Builder Notation:

${x | x > 0}$

Interval Notation:

$(0, \infty)$

Set-Builder Notation:

${x | x > 0}$

Step 3

Create intervals around the $x$ -values where the second derivative is zero or undefined.

$(0, \infty)$

Step 4

Substitute any number from the interval $(0, \infty)$ into the second derivative and evaluate to determine the concavity.

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Step 4.1

Replace the variable $x$ with $2$ in the expression.

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

Step 4.2

The final answer is $\frac{1}{2}$ .

$\frac{1}{2}$

Step 4.3

The graph is concave up on the interval $(0, \infty)$ because $f'' (2)$ is positive.

Concave up on $(0, \infty)$ since $f'' (x)$ is positive

Step 5