Calculus Examples

$y = x^{2} - 8 \ln (x)$

Step 1

Find the first derivative.

Tap for more steps...

Step 1.1

Find the first derivative.

Tap for more steps...

Step 1.1.1

Differentiate.

Tap for more steps...

Step 1.1.1.1

By the Sum Rule, the derivative of $x^{2} - 8 \ln (x)$ with respect to $x$ is $\frac{d}{d x} [x^{2}] + \frac{d}{d x} [- 8 \ln (x)]$ .

$\frac{d}{d x} [x^{2}] + \frac{d}{d x} [- 8 \ln (x)]$

Step 1.1.1.2

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 2$ .

$2 x + \frac{d}{d x} [- 8 \ln (x)]$

Step 1.1.2

Evaluate $\frac{d}{d x} [- 8 \ln (x)]$ .

Tap for more steps...

Step 1.1.2.1

Since $- 8$ is constant with respect to $x$ , the derivative of $- 8 \ln (x)$ with respect to $x$ is $- 8 \frac{d}{d x} [\ln (x)]$ .

$2 x - 8 \frac{d}{d x} [\ln (x)]$

Step 1.1.2.2

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

$2 x - 8 \frac{1}{x}$

Step 1.1.2.3

Combine $- 8$ and $\frac{1}{x}$ .

$2 x + \frac{- 8}{x}$

Step 1.1.2.4

Move the negative in front of the fraction.

$f' (x) = 2 x - \frac{8}{x}$

Step 1.2

The first derivative of $f (x)$ with respect to $x$ is $2 x - \frac{8}{x}$ .

$2 x - \frac{8}{x}$

Step 2

Set the first derivative equal to $0$ then solve the equation $2 x - \frac{8}{x} = 0$ .

Tap for more steps...

Step 2.1

Set the first derivative equal to $0$ .

$2 x - \frac{8}{x} = 0$

Step 2.2

Find the LCD of the terms in the equation.

Tap for more steps...

Step 2.2.1

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

$1, x, 1$

Step 2.2.2

The LCM of one and any expression is the expression.

$x$

Step 2.3

Multiply each term in $2 x - \frac{8}{x} = 0$ by $x$ to eliminate the fractions.

Tap for more steps...

Step 2.3.1

Multiply each term in $2 x - \frac{8}{x} = 0$ by $x$ .

$2 x \cdot x - \frac{8}{x} x = 0 x$

Step 2.3.2

Simplify the left side.

Tap for more steps...

Step 2.3.2.1

Simplify each term.

Tap for more steps...

Step 2.3.2.1.1

Multiply $x$ by $x$ by adding the exponents.

Tap for more steps...

Step 2.3.2.1.1.1

Move $x$ .

$2 (x \cdot x) - \frac{8}{x} x = 0 x$

Step 2.3.2.1.1.2

Multiply $x$ by $x$ .

$2 x^{2} - \frac{8}{x} x = 0 x$

Step 2.3.2.1.2

Cancel the common factor of $x$ .

Tap for more steps...

Step 2.3.2.1.2.1

Move the leading negative in $- \frac{8}{x}$ into the numerator.

$2 x^{2} + \frac{- 8}{x} x = 0 x$

Step 2.3.2.1.2.2

Cancel the common factor.

$2 x^{2} + \frac{- 8}{x} x = 0 x$

Step 2.3.2.1.2.3

Rewrite the expression.

$2 x^{2} - 8 = 0 x$

Step 2.3.3

Simplify the right side.

Tap for more steps...

Step 2.3.3.1

Multiply $0$ by $x$ .

$2 x^{2} - 8 = 0$

Step 2.4

Solve the equation.

Tap for more steps...

Step 2.4.1

Add $8$ to both sides of the equation.

$2 x^{2} = 8$

Step 2.4.2

Divide each term in $2 x^{2} = 8$ by $2$ and simplify.

Tap for more steps...

Step 2.4.2.1

Divide each term in $2 x^{2} = 8$ by $2$ .

$\frac{2 x^{2}}{2} = \frac{8}{2}$

Step 2.4.2.2

Simplify the left side.

Tap for more steps...

Step 2.4.2.2.1

Cancel the common factor of $2$ .

Tap for more steps...

Step 2.4.2.2.1.1

Cancel the common factor.

$\frac{2 x^{2}}{2} = \frac{8}{2}$

Step 2.4.2.2.1.2

Divide $x^{2}$ by $1$ .

$x^{2} = \frac{8}{2}$

Step 2.4.2.3

Simplify the right side.

Tap for more steps...

Step 2.4.2.3.1

Divide $8$ by $2$ .

$x^{2} = 4$

Step 2.4.3

Take the specified root of both sides of the equation to eliminate the exponent on the left side.

$x = \pm \sqrt{4}$

Step 2.4.4

Simplify $\pm \sqrt{4}$ .

Tap for more steps...

Step 2.4.4.1

Rewrite $4$ as $2^{2}$ .

$x = \pm \sqrt{2^{2}}$

Step 2.4.4.2

Pull terms out from under the radical, assuming positive real numbers.

$x = \pm 2$

Step 2.4.5

The complete solution is the result of both the positive and negative portions of the solution.

Tap for more steps...

Step 2.4.5.1

First, use the positive value of the $\pm$ to find the first solution.

$x = 2$

Step 2.4.5.2

Next, use the negative value of the $\pm$ to find the second solution.

$x = - 2$

Step 2.4.5.3

The complete solution is the result of both the positive and negative portions of the solution.

$x = 2, - 2$

Step 3

Find the values where the derivative is undefined.

Tap for more steps...

Step 3.1

Set the denominator in $\frac{8}{x}$ equal to $0$ to find where the expression is undefined.

$x = 0$

Step 4

Evaluate $x^{2} - 8 \ln (x)$ at each $x$ value where the derivative is $0$ or undefined.

Tap for more steps...

Step 4.1

Evaluate at $x = 2$ .

Tap for more steps...

Step 4.1.1

Substitute $2$ for $x$ .

${(2)}^{2} - 8 \ln (2)$

Step 4.1.2

Simplify each term.

Tap for more steps...

Step 4.1.2.1

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

$4 - 8 \ln (2)$

Step 4.1.2.2

Simplify $- 8 \ln (2)$ by moving $8$ inside the logarithm.

$4 - \ln (2^{8})$

Step 4.1.2.3

Raise $2$ to the power of $8$ .

$4 - \ln (256)$

Step 4.2

Evaluate at $x = - 2$ .

Tap for more steps...

Step 4.2.1

Substitute $- 2$ for $x$ .

${(- 2)}^{2} - 8 \ln (- 2)$

Step 4.2.2

The natural logarithm of a negative number is undefined.

Undefined

Step 4.3

Evaluate at $x = 0$ .

Tap for more steps...

Step 4.3.1

Substitute $0$ for $x$ .

${(0)}^{2} - 8 \ln (0)$

Step 4.3.2

The natural logarithm of zero is undefined.

Undefined

Step 4.4

List all of the points.

$(2, 4 - \ln (256))$

Step 5