Calculus Examples

$\frac{x - 1}{x - 6}$

Step 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 $f (x) = x - 1$ and $g (x) = x - 6$ .

$\frac{(x - 6) \frac{d}{d x} [x - 1] - (x - 1) \frac{d}{d x} [x - 6]}{{(x - 6)}^{2}}$

Step 2

Differentiate.

Tap for more steps...

Step 2.1

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

$\frac{(x - 6) (\frac{d}{d x} [x] + \frac{d}{d x} [- 1]) - (x - 1) \frac{d}{d x} [x - 6]}{{(x - 6)}^{2}}$

Step 2.2

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

$\frac{(x - 6) (1 + \frac{d}{d x} [- 1]) - (x - 1) \frac{d}{d x} [x - 6]}{{(x - 6)}^{2}}$

Step 2.3

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

$\frac{(x - 6) (1 + 0) - (x - 1) \frac{d}{d x} [x - 6]}{{(x - 6)}^{2}}$

Step 2.4

Simplify the expression.

Tap for more steps...

Step 2.4.1

Add $1$ and $0$ .

$\frac{(x - 6) \cdot 1 - (x - 1) \frac{d}{d x} [x - 6]}{{(x - 6)}^{2}}$

Step 2.4.2

Multiply $x - 6$ by $1$ .

$\frac{x - 6 - (x - 1) \frac{d}{d x} [x - 6]}{{(x - 6)}^{2}}$

Step 2.5

By the Sum Rule, the derivative of $x - 6$ with respect to $x$ is $\frac{d}{d x} [x] + \frac{d}{d x} [- 6]$ .

$\frac{x - 6 - (x - 1) (\frac{d}{d x} [x] + \frac{d}{d x} [- 6])}{{(x - 6)}^{2}}$

Step 2.6

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

$\frac{x - 6 - (x - 1) (1 + \frac{d}{d x} [- 6])}{{(x - 6)}^{2}}$

Step 2.7

Since $- 6$ is constant with respect to $x$ , the derivative of $- 6$ with respect to $x$ is $0$ .

$\frac{x - 6 - (x - 1) (1 + 0)}{{(x - 6)}^{2}}$

Step 2.8

Simplify the expression.

Tap for more steps...

Step 2.8.1

Add $1$ and $0$ .

$\frac{x - 6 - (x - 1) \cdot 1}{{(x - 6)}^{2}}$

Step 2.8.2

Multiply $- 1$ by $1$ .

$\frac{x - 6 - (x - 1)}{{(x - 6)}^{2}}$

Step 3

Simplify.

Tap for more steps...

Step 3.1

Apply the distributive property.

$\frac{x - 6 - x - - 1}{{(x - 6)}^{2}}$

Step 3.2

Simplify the numerator.

Tap for more steps...

Step 3.2.1

Combine the opposite terms in $x - 6 - x - - 1$ .

Tap for more steps...

Step 3.2.1.1

Subtract $x$ from $x$ .

$\frac{0 - 6 - - 1}{{(x - 6)}^{2}}$

Step 3.2.1.2

Subtract $6$ from $0$ .

$\frac{- 6 - - 1}{{(x - 6)}^{2}}$

Step 3.2.2

Multiply $- 1$ by $- 1$ .

$\frac{- 6 + 1}{{(x - 6)}^{2}}$

Step 3.2.3

Add $- 6$ and $1$ .

$\frac{- 5}{{(x - 6)}^{2}}$

Step 3.3

Move the negative in front of the fraction.

$- \frac{5}{{(x - 6)}^{2}}$