Calculus Examples

Let $f (x) = \frac{x + 1}{x - 3}$

Step 1

Find the first derivative.

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Step 1.1

Find the first derivative.

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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 $f (x) = x + 1$ and $g (x) = x - 3$ .

$\frac{(x - 3) \frac{d}{d x} [x + 1] - (x + 1) \frac{d}{d x} [x - 3]}{{(x - 3)}^{2}}$

Step 1.1.2

Differentiate.

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Step 1.1.2.1

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

$\frac{(x - 3) (\frac{d}{d x} [x] + \frac{d}{d x} [1]) - (x + 1) \frac{d}{d x} [x - 3]}{{(x - 3)}^{2}}$

Step 1.1.2.2

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

$\frac{(x - 3) (1 + \frac{d}{d x} [1]) - (x + 1) \frac{d}{d x} [x - 3]}{{(x - 3)}^{2}}$

Step 1.1.2.3

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

$\frac{(x - 3) (1 + 0) - (x + 1) \frac{d}{d x} [x - 3]}{{(x - 3)}^{2}}$

Step 1.1.2.4

Simplify the expression.

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Step 1.1.2.4.1

Add $1$ and $0$ .

$\frac{(x - 3) \cdot 1 - (x + 1) \frac{d}{d x} [x - 3]}{{(x - 3)}^{2}}$

Step 1.1.2.4.2

Multiply $x - 3$ by $1$ .

$\frac{x - 3 - (x + 1) \frac{d}{d x} [x - 3]}{{(x - 3)}^{2}}$

Step 1.1.2.5

By the Sum Rule, the derivative of $x - 3$ with respect to $x$ is $\frac{d}{d x} [x] + \frac{d}{d x} [- 3]$ .

$\frac{x - 3 - (x + 1) (\frac{d}{d x} [x] + \frac{d}{d x} [- 3])}{{(x - 3)}^{2}}$

Step 1.1.2.6

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

$\frac{x - 3 - (x + 1) (1 + \frac{d}{d x} [- 3])}{{(x - 3)}^{2}}$

Step 1.1.2.7

Since $- 3$ is constant with respect to $x$ , the derivative of $- 3$ with respect to $x$ is $0$ .

$\frac{x - 3 - (x + 1) (1 + 0)}{{(x - 3)}^{2}}$

Step 1.1.2.8

Simplify the expression.

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Step 1.1.2.8.1

Add $1$ and $0$ .

$\frac{x - 3 - (x + 1) \cdot 1}{{(x - 3)}^{2}}$

Step 1.1.2.8.2

Multiply $- 1$ by $1$ .

$\frac{x - 3 - (x + 1)}{{(x - 3)}^{2}}$

Step 1.1.3

Simplify.

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Step 1.1.3.1

Apply the distributive property.

$\frac{x - 3 - x - 1 \cdot 1}{{(x - 3)}^{2}}$

Step 1.1.3.2

Simplify the numerator.

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Step 1.1.3.2.1

Combine the opposite terms in $x - 3 - x - 1 \cdot 1$ .

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Step 1.1.3.2.1.1

Subtract $x$ from $x$ .

$\frac{0 - 3 - 1 \cdot 1}{{(x - 3)}^{2}}$

Step 1.1.3.2.1.2

Subtract $3$ from $0$ .

$\frac{- 3 - 1 \cdot 1}{{(x - 3)}^{2}}$

Step 1.1.3.2.2

Multiply $- 1$ by $1$ .

$\frac{- 3 - 1}{{(x - 3)}^{2}}$

Step 1.1.3.2.3

Subtract $1$ from $- 3$ .

$\frac{- 4}{{(x - 3)}^{2}}$

Step 1.1.3.3

Move the negative in front of the fraction.

$f' (x) = - \frac{4}{{(x - 3)}^{2}}$

Step 1.2

The first derivative of $f (x)$ with respect to $x$ is $- \frac{4}{{(x - 3)}^{2}}$ .

$- \frac{4}{{(x - 3)}^{2}}$

Step 2

Set the first derivative equal to $0$ then solve the equation $- \frac{4}{{(x - 3)}^{2}} = 0$ .

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Step 2.1

Set the first derivative equal to $0$ .

$- \frac{4}{{(x - 3)}^{2}} = 0$

Step 2.2

Set the numerator equal to zero.

$4 = 0$

Step 2.3

Since $4 \neq 0$ , there are no solutions.

No solution

Step 3

Find the values where the derivative is undefined.

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Step 3.1

Set the denominator in $\frac{4}{{(x - 3)}^{2}}$ equal to $0$ to find where the expression is undefined.

${(x - 3)}^{2} = 0$

Step 3.2

Solve for $x$ .

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Step 3.2.1

Set the $x - 3$ equal to $0$ .

$x - 3 = 0$

Step 3.2.2

Add $3$ to both sides of the equation.

$x = 3$

Step 4

Evaluate $\frac{x + 1}{x - 3}$ at each $x$ value where the derivative is $0$ or undefined.

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Step 4.1

Evaluate at $x = 3$ .

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Step 4.1.1

Substitute $3$ for $x$ .

$\frac{(3) + 1}{(3) - 3}$

Step 4.1.2

Simplify.

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Step 4.1.2.1

Subtract $3$ from $3$ .

$\frac{(3) + 1}{0}$

Step 4.1.2.2

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