Calculus Examples

$f (x) = \frac{64 x^{2} - 25}{x}$

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 using the Quotient Rule which states that $\frac{d}{d x} [\frac{f (x)}{g (x)}]$ is $\frac{g (x) \frac{d}{d x} [f (x)] - f (x) \frac{d}{d x} [g (x)]}{{g (x)}^{2}}$ where $f (x) = 64 x^{2} - 25$ and $g (x) = x$ .

$\frac{x \frac{d}{d x} [64 x^{2} - 25] - (64 x^{2} - 25) \frac{d}{d x} [x]}{x^{2}}$

Step 1.1.2

Differentiate.

Tap for more steps...

Step 1.1.2.1

By the Sum Rule, the derivative of $64 x^{2} - 25$ with respect to $x$ is $\frac{d}{d x} [64 x^{2}] + \frac{d}{d x} [- 25]$ .

$\frac{x (\frac{d}{d x} [64 x^{2}] + \frac{d}{d x} [- 25]) - (64 x^{2} - 25) \frac{d}{d x} [x]}{x^{2}}$

Step 1.1.2.2

Since $64$ is constant with respect to $x$ , the derivative of $64 x^{2}$ with respect to $x$ is $64 \frac{d}{d x} [x^{2}]$ .

$\frac{x (64 \frac{d}{d x} [x^{2}] + \frac{d}{d x} [- 25]) - (64 x^{2} - 25) \frac{d}{d x} [x]}{x^{2}}$

Step 1.1.2.3

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

$\frac{x (64 (2 x) + \frac{d}{d x} [- 25]) - (64 x^{2} - 25) \frac{d}{d x} [x]}{x^{2}}$

Step 1.1.2.4

Multiply $2$ by $64$ .

$\frac{x (128 x + \frac{d}{d x} [- 25]) - (64 x^{2} - 25) \frac{d}{d x} [x]}{x^{2}}$

Step 1.1.2.5

Since $- 25$ is constant with respect to $x$ , the derivative of $- 25$ with respect to $x$ is $0$ .

$\frac{x (128 x + 0) - (64 x^{2} - 25) \frac{d}{d x} [x]}{x^{2}}$

Step 1.1.2.6

Add $128 x$ and $0$ .

$\frac{x (128 x) - (64 x^{2} - 25) \frac{d}{d x} [x]}{x^{2}}$

Step 1.1.3

Raise $x$ to the power of $1$ .

$\frac{128 (x^{1} x) - (64 x^{2} - 25) \frac{d}{d x} [x]}{x^{2}}$

Step 1.1.4

Raise $x$ to the power of $1$ .

$\frac{128 (x^{1} x^{1}) - (64 x^{2} - 25) \frac{d}{d x} [x]}{x^{2}}$

Step 1.1.5

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$\frac{128 x^{1 + 1} - (64 x^{2} - 25) \frac{d}{d x} [x]}{x^{2}}$

Step 1.1.6

Add $1$ and $1$ .

$\frac{128 x^{2} - (64 x^{2} - 25) \frac{d}{d x} [x]}{x^{2}}$

Step 1.1.7

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

$\frac{128 x^{2} - (64 x^{2} - 25) \cdot 1}{x^{2}}$

Step 1.1.8

Multiply $- 1$ by $1$ .

$\frac{128 x^{2} - (64 x^{2} - 25)}{x^{2}}$

Step 1.1.9

Simplify.

Tap for more steps...

Step 1.1.9.1

Apply the distributive property.

$\frac{128 x^{2} - (64 x^{2}) - - 25}{x^{2}}$

Step 1.1.9.2

Simplify the numerator.

Tap for more steps...

Step 1.1.9.2.1

Simplify each term.

Tap for more steps...

Step 1.1.9.2.1.1

Multiply $64$ by $- 1$ .

$\frac{128 x^{2} - 64 x^{2} - - 25}{x^{2}}$

Step 1.1.9.2.1.2

Multiply $- 1$ by $- 25$ .

$\frac{128 x^{2} - 64 x^{2} + 25}{x^{2}}$

Step 1.1.9.2.2

Subtract $64 x^{2}$ from $128 x^{2}$ .

$f' (x) = \frac{64 x^{2} + 25}{x^{2}}$

Step 1.2

The first derivative of $f (x)$ with respect to $x$ is $\frac{64 x^{2} + 25}{x^{2}}$ .

$\frac{64 x^{2} + 25}{x^{2}}$

Step 2

Set the first derivative equal to $0$ then solve the equation $\frac{64 x^{2} + 25}{x^{2}} = 0$ .

Tap for more steps...

Step 2.1

Set the first derivative equal to $0$ .

$\frac{64 x^{2} + 25}{x^{2}} = 0$

Step 2.2

Set the numerator equal to zero.

$64 x^{2} + 25 = 0$

Step 2.3

Solve the equation for $x$ .

Tap for more steps...

Step 2.3.1

Subtract $25$ from both sides of the equation.

$64 x^{2} = - 25$

Step 2.3.2

Divide each term in $64 x^{2} = - 25$ by $64$ and simplify.

Tap for more steps...

Step 2.3.2.1

Divide each term in $64 x^{2} = - 25$ by $64$ .

$\frac{64 x^{2}}{64} = \frac{- 25}{64}$

Step 2.3.2.2

Simplify the left side.

Tap for more steps...

Step 2.3.2.2.1

Cancel the common factor of $64$ .

Tap for more steps...

Step 2.3.2.2.1.1

Cancel the common factor.

$\frac{64 x^{2}}{64} = \frac{- 25}{64}$

Step 2.3.2.2.1.2

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

$x^{2} = \frac{- 25}{64}$

Step 2.3.2.3

Simplify the right side.

Tap for more steps...

Step 2.3.2.3.1

Move the negative in front of the fraction.

$x^{2} = - \frac{25}{64}$

Step 2.3.3

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

$x = \pm \sqrt{- \frac{25}{64}}$

Step 2.3.4

Simplify $\pm \sqrt{- \frac{25}{64}}$ .

Tap for more steps...

Step 2.3.4.1

Rewrite $- \frac{25}{64}$ as ${(\frac{5 i}{8})}^{2}$ .

$x = \pm \sqrt{{(\frac{5 i}{8})}^{2}}$

Step 2.3.4.2

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

$x = \pm \frac{5 i}{8}$

Step 2.3.5

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

Tap for more steps...

Step 2.3.5.1

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

$x = \frac{5 i}{8}$

Step 2.3.5.2

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

$x = - \frac{5 i}{8}$

Step 2.3.5.3

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

$x = \frac{5 i}{8}, - \frac{5 i}{8}$

Step 3

Find the values where the derivative is undefined.

Tap for more steps...

Step 3.1

Set the denominator in $\frac{64 x^{2} + 25}{x^{2}}$ equal to $0$ to find where the expression is undefined.

$x^{2} = 0$

Step 3.2

Solve for $x$ .

Tap for more steps...

Step 3.2.1

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

$x = \pm \sqrt{0}$

Step 3.2.2

Simplify $\pm \sqrt{0}$ .

Tap for more steps...

Step 3.2.2.1

Rewrite $0$ as $0^{2}$ .

$x = \pm \sqrt{0^{2}}$

Step 3.2.2.2

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

$x = \pm 0$

Step 3.2.2.3

Plus or minus $0$ is $0$ .

$x = 0$

Step 4

Evaluate $\frac{64 x^{2} - 25}{x}$ at each $x$ value where the derivative is $0$ or undefined.

Tap for more steps...

Step 4.1

Evaluate at $x = 0$ .

Tap for more steps...

Step 4.1.1

Substitute $0$ for $x$ .

$\frac{64 {(0)}^{2} - 25}{0}$

Step 4.1.2

The expression contains a division by $0$ . The expression is undefined.

Undefined

Step 5

There are no values of $x$ in the domain of the original problem where the derivative is $0$ or undefined.

No critical points found