Calculus Examples

$f (x) = \frac{1}{x^{2} - 2 x + 8}$

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

Rewrite $\frac{1}{x^{2} - 2 x + 8}$ as ${(x^{2} - 2 x + 8)}^{- 1}$ .

$\frac{d}{d x} [{(x^{2} - 2 x + 8)}^{- 1}]$

Step 1.1.2

Differentiate using the chain rule, which states that $\frac{d}{d x} [f (g (x))]$ is $f' (g (x)) g' (x)$ where $f (x) = x^{- 1}$ and $g (x) = x^{2} - 2 x + 8$ .

Tap for more steps...

Step 1.1.2.1

To apply the Chain Rule, set $u$ as $x^{2} - 2 x + 8$ .

$\frac{d}{d u} [u^{- 1}] \frac{d}{d x} [x^{2} - 2 x + 8]$

Step 1.1.2.2

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

$- u^{- 2} \frac{d}{d x} [x^{2} - 2 x + 8]$

Step 1.1.2.3

Replace all occurrences of $u$ with $x^{2} - 2 x + 8$ .

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

Step 1.1.3

Differentiate.

Tap for more steps...

Step 1.1.3.1

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

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

Step 1.1.3.2

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

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

Step 1.1.3.3

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

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

Step 1.1.3.4

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

$- {(x^{2} - 2 x + 8)}^{- 2} (2 x - 2 \cdot 1 + \frac{d}{d x} [8])$

Step 1.1.3.5

Multiply $- 2$ by $1$ .

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

Step 1.1.3.6

Since $8$ is constant with respect to $x$ , the derivative of $8$ with respect to $x$ is $0$ .

$- {(x^{2} - 2 x + 8)}^{- 2} (2 x - 2 + 0)$

Step 1.1.3.7

Add $2 x - 2$ and $0$ .

$- {(x^{2} - 2 x + 8)}^{- 2} (2 x - 2)$

Step 1.1.4

Simplify.

Tap for more steps...

Step 1.1.4.1

Rewrite the expression using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$- \frac{1}{{(x^{2} - 2 x + 8)}^{2}} (2 x - 2)$

Step 1.1.4.2

Reorder the factors of $- \frac{1}{{(x^{2} - 2 x + 8)}^{2}} (2 x - 2)$ .

$- (2 x - 2) \frac{1}{{(x^{2} - 2 x + 8)}^{2}}$

Step 1.1.4.3

Apply the distributive property.

$(- (2 x) - - 2) \frac{1}{{(x^{2} - 2 x + 8)}^{2}}$

Step 1.1.4.4

Multiply $2$ by $- 1$ .

$(- 2 x - - 2) \frac{1}{{(x^{2} - 2 x + 8)}^{2}}$

Step 1.1.4.5

Multiply $- 1$ by $- 2$ .

$(- 2 x + 2) \frac{1}{{(x^{2} - 2 x + 8)}^{2}}$

Step 1.1.4.6

Multiply $- 2 x + 2$ by $\frac{1}{{(x^{2} - 2 x + 8)}^{2}}$ .

$\frac{- 2 x + 2}{{(x^{2} - 2 x + 8)}^{2}}$

Step 1.1.4.7

Factor $2$ out of $- 2 x + 2$ .

Tap for more steps...

Step 1.1.4.7.1

Factor $2$ out of $- 2 x$ .

$\frac{2 (- x) + 2}{{(x^{2} - 2 x + 8)}^{2}}$

Step 1.1.4.7.2

Factor $2$ out of $2$ .

$\frac{2 (- x) + 2 (1)}{{(x^{2} - 2 x + 8)}^{2}}$

Step 1.1.4.7.3

Factor $2$ out of $2 (- x) + 2 (1)$ .

$\frac{2 (- x + 1)}{{(x^{2} - 2 x + 8)}^{2}}$

Step 1.1.4.8

Factor $- 1$ out of $- x$ .

$\frac{2 (- (x) + 1)}{{(x^{2} - 2 x + 8)}^{2}}$

Step 1.1.4.9

Rewrite $1$ as $- 1 (- 1)$ .

$\frac{2 (- (x) - 1 (- 1))}{{(x^{2} - 2 x + 8)}^{2}}$

Step 1.1.4.10

Factor $- 1$ out of $- (x) - 1 (- 1)$ .

$\frac{2 (- (x - 1))}{{(x^{2} - 2 x + 8)}^{2}}$

Step 1.1.4.11

Rewrite $- (x - 1)$ as $- 1 (x - 1)$ .

$\frac{2 (- 1 (x - 1))}{{(x^{2} - 2 x + 8)}^{2}}$

Step 1.1.4.12

Move the negative in front of the fraction.

$f' (x) = - \frac{2 (x - 1)}{{(x^{2} - 2 x + 8)}^{2}}$

Step 1.2

The first derivative of $f (x)$ with respect to $x$ is $- \frac{2 (x - 1)}{{(x^{2} - 2 x + 8)}^{2}}$ .

$- \frac{2 (x - 1)}{{(x^{2} - 2 x + 8)}^{2}}$

Step 2

Set the first derivative equal to $0$ then solve the equation $- \frac{2 (x - 1)}{{(x^{2} - 2 x + 8)}^{2}} = 0$ .

Tap for more steps...

Step 2.1

Set the first derivative equal to $0$ .

$- \frac{2 (x - 1)}{{(x^{2} - 2 x + 8)}^{2}} = 0$

Step 2.2

Set the numerator equal to zero.

$2 (x - 1) = 0$

Step 2.3

Solve the equation for $x$ .

Tap for more steps...

Step 2.3.1

Divide each term in $2 (x - 1) = 0$ by $2$ and simplify.

Tap for more steps...

Step 2.3.1.1

Divide each term in $2 (x - 1) = 0$ by $2$ .

$\frac{2 (x - 1)}{2} = \frac{0}{2}$

Step 2.3.1.2

Simplify the left side.

Tap for more steps...

Step 2.3.1.2.1

Cancel the common factor of $2$ .

Tap for more steps...

Step 2.3.1.2.1.1

Cancel the common factor.

$\frac{2 (x - 1)}{2} = \frac{0}{2}$

Step 2.3.1.2.1.2

Divide $x - 1$ by $1$ .

$x - 1 = \frac{0}{2}$

Step 2.3.1.3

Simplify the right side.

Tap for more steps...

Step 2.3.1.3.1

Divide $0$ by $2$ .

$x - 1 = 0$

Step 2.3.2

Add $1$ to both sides of the equation.

$x = 1$

Step 3

Find the values where the derivative is undefined.

Tap for more steps...

Step 3.1

The domain of the expression is all real numbers except where the expression is undefined. In this case, there is no real number that makes the expression undefined.

Step 4

Evaluate $\frac{1}{x^{2} - 2 x + 8}$ at each $x$ value where the derivative is $0$ or undefined.

Tap for more steps...

Step 4.1

Evaluate at $x = 1$ .

Tap for more steps...

Step 4.1.1

Substitute $1$ for $x$ .

$\frac{1}{{(1)}^{2} - 2 \cdot 1 + 8}$

Step 4.1.2

Simplify the denominator.

Tap for more steps...

Step 4.1.2.1

One to any power is one.

$\frac{1}{1 - 2 \cdot 1 + 8}$

Step 4.1.2.2

Multiply $- 2$ by $1$ .

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

Step 4.1.2.3

Subtract $2$ from $1$ .

$\frac{1}{- 1 + 8}$

Step 4.1.2.4

Add $- 1$ and $8$ .

$\frac{1}{7}$

Step 4.2

List all of the points.

$(1, \frac{1}{7})$

Step 5