Calculus Examples

$f (x) = \frac{7 - 2 x}{5 + x}$

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 $f (x) = 7 - 2 x$ and $g (x) = 5 + x$ .

$\frac{(5 + x) \frac{d}{d x} [7 - 2 x] - (7 - 2 x) \frac{d}{d x} [5 + x]}{{(5 + x)}^{2}}$

Step 1.1.2

Differentiate.

Tap for more steps...

Step 1.1.2.1

By the Sum Rule, the derivative of $7 - 2 x$ with respect to $x$ is $\frac{d}{d x} [7] + \frac{d}{d x} [- 2 x]$ .

$\frac{(5 + x) (\frac{d}{d x} [7] + \frac{d}{d x} [- 2 x]) - (7 - 2 x) \frac{d}{d x} [5 + x]}{{(5 + x)}^{2}}$

Step 1.1.2.2

Since $7$ is constant with respect to $x$ , the derivative of $7$ with respect to $x$ is $0$ .

$\frac{(5 + x) (0 + \frac{d}{d x} [- 2 x]) - (7 - 2 x) \frac{d}{d x} [5 + x]}{{(5 + x)}^{2}}$

Step 1.1.2.3

Add $0$ and $\frac{d}{d x} [- 2 x]$ .

$\frac{(5 + x) \frac{d}{d x} [- 2 x] - (7 - 2 x) \frac{d}{d x} [5 + x]}{{(5 + x)}^{2}}$

Step 1.1.2.4

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

$\frac{(5 + x) (- 2 \frac{d}{d x} [x]) - (7 - 2 x) \frac{d}{d x} [5 + x]}{{(5 + x)}^{2}}$

Step 1.1.2.5

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

$\frac{(5 + x) (- 2 \cdot 1) - (7 - 2 x) \frac{d}{d x} [5 + x]}{{(5 + x)}^{2}}$

Step 1.1.2.6

Simplify the expression.

Tap for more steps...

Step 1.1.2.6.1

Multiply $- 2$ by $1$ .

$\frac{(5 + x) \cdot - 2 - (7 - 2 x) \frac{d}{d x} [5 + x]}{{(5 + x)}^{2}}$

Step 1.1.2.6.2

Move $- 2$ to the left of $5 + x$ .

$\frac{- 2 \cdot (5 + x) - (7 - 2 x) \frac{d}{d x} [5 + x]}{{(5 + x)}^{2}}$

Step 1.1.2.7

By the Sum Rule, the derivative of $5 + x$ with respect to $x$ is $\frac{d}{d x} [5] + \frac{d}{d x} [x]$ .

$\frac{- 2 (5 + x) - (7 - 2 x) (\frac{d}{d x} [5] + \frac{d}{d x} [x])}{{(5 + x)}^{2}}$

Step 1.1.2.8

Since $5$ is constant with respect to $x$ , the derivative of $5$ with respect to $x$ is $0$ .

$\frac{- 2 (5 + x) - (7 - 2 x) (0 + \frac{d}{d x} [x])}{{(5 + x)}^{2}}$

Step 1.1.2.9

Add $0$ and $\frac{d}{d x} [x]$ .

$\frac{- 2 (5 + x) - (7 - 2 x) \frac{d}{d x} [x]}{{(5 + x)}^{2}}$

Step 1.1.2.10

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

$\frac{- 2 (5 + x) - (7 - 2 x) \cdot 1}{{(5 + x)}^{2}}$

Step 1.1.2.11

Multiply $- 1$ by $1$ .

$\frac{- 2 (5 + x) - (7 - 2 x)}{{(5 + x)}^{2}}$

Step 1.1.3

Simplify.

Tap for more steps...

Step 1.1.3.1

Apply the distributive property.

$\frac{- 2 \cdot 5 - 2 x - (7 - 2 x)}{{(5 + x)}^{2}}$

Step 1.1.3.2

Apply the distributive property.

$\frac{- 2 \cdot 5 - 2 x - 1 \cdot 7 - (- 2 x)}{{(5 + x)}^{2}}$

Step 1.1.3.3

Simplify the numerator.

Tap for more steps...

Step 1.1.3.3.1

Simplify each term.

Tap for more steps...

Step 1.1.3.3.1.1

Multiply $- 2$ by $5$ .

$\frac{- 10 - 2 x - 1 \cdot 7 - (- 2 x)}{{(5 + x)}^{2}}$

Step 1.1.3.3.1.2

Multiply $- 1$ by $7$ .

$\frac{- 10 - 2 x - 7 - (- 2 x)}{{(5 + x)}^{2}}$

Step 1.1.3.3.1.3

Multiply $- 2$ by $- 1$ .

$\frac{- 10 - 2 x - 7 + 2 x}{{(5 + x)}^{2}}$

Step 1.1.3.3.2

Combine the opposite terms in $- 10 - 2 x - 7 + 2 x$ .

Tap for more steps...

Step 1.1.3.3.2.1

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

$\frac{- 10 + 0 - 7}{{(5 + x)}^{2}}$

Step 1.1.3.3.2.2

Add $- 10$ and $0$ .

$\frac{- 10 - 7}{{(5 + x)}^{2}}$

Step 1.1.3.3.3

Subtract $7$ from $- 10$ .

$\frac{- 17}{{(5 + x)}^{2}}$

Step 1.1.3.4

Move the negative in front of the fraction.

$f' (x) = - \frac{17}{{(5 + x)}^{2}}$

Step 1.2

The first derivative of $f (x)$ with respect to $x$ is $- \frac{17}{{(5 + x)}^{2}}$ .

$- \frac{17}{{(5 + x)}^{2}}$

Step 2

Set the first derivative equal to $0$ then solve the equation $- \frac{17}{{(5 + x)}^{2}} = 0$ .

Tap for more steps...

Step 2.1

Set the first derivative equal to $0$ .

$- \frac{17}{{(5 + x)}^{2}} = 0$

Step 2.2

Set the numerator equal to zero.

$17 = 0$

Step 2.3

Since $17 \neq 0$ , there are no solutions.

No solution

Step 3

Find the values where the derivative is undefined.

Tap for more steps...

Step 3.1

Set the denominator in $\frac{17}{{(5 + x)}^{2}}$ equal to $0$ to find where the expression is undefined.

${(5 + x)}^{2} = 0$

Step 3.2

Solve for $x$ .

Tap for more steps...

Step 3.2.1

Set the $5 + x$ equal to $0$ .

$5 + x = 0$

Step 3.2.2

Subtract $5$ from both sides of the equation.

$x = - 5$

Step 4

Evaluate $\frac{7 - 2 x}{5 + x}$ at each $x$ value where the derivative is $0$ or undefined.

Tap for more steps...

Step 4.1

Evaluate at $x = - 5$ .

Tap for more steps...

Step 4.1.1

Substitute $- 5$ for $x$ .

$\frac{7 - 2 \cdot - 5}{5 - 5}$

Step 4.1.2

Simplify.

Tap for more steps...

Step 4.1.2.1

Remove parentheses.

$\frac{7 - 2 \cdot - 5}{5 - 5}$

Step 4.1.2.2

Subtract $5$ from $5$ .

$\frac{7 - 2 \cdot - 5}{0}$

Step 4.1.2.3

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