Calculus Examples

$y = \frac{1}{x - 2} - 3$

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

By the Sum Rule, the derivative of $\frac{1}{x - 2} - 3$ with respect to $x$ is $\frac{d}{d x} [\frac{1}{x - 2}] + \frac{d}{d x} [- 3]$ .

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

Step 1.1.2

Evaluate $\frac{d}{d x} [\frac{1}{x - 2}]$ .

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Step 1.1.2.1

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

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

Step 1.1.2.2

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 $g (x) = x - 2$ .

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Step 1.1.2.2.1

To apply the Chain Rule, set $u$ as $x - 2$ .

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

Step 1.1.2.2.2

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

$- u^{- 2} \frac{d}{d x} [x - 2] + \frac{d}{d x} [- 3]$

Step 1.1.2.2.3

Replace all occurrences of $u$ with $x - 2$ .

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

Step 1.1.2.3

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

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

Step 1.1.2.4

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

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

Step 1.1.2.5

Since $- 2$ is constant with respect to $x$ , the derivative of $- 2$ with respect to $x$ is $0$ .

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

Step 1.1.2.6

Add $1$ and $0$ .

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

Step 1.1.2.7

Multiply $- 1$ by $1$ .

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

Step 1.1.3

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

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

Step 1.1.4

Simplify.

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Step 1.1.4.1

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

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

Step 1.1.4.2

Add $- \frac{1}{{(x - 2)}^{2}}$ and $0$ .

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

Step 1.2

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

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

Step 2

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

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Step 2.1

Set the first derivative equal to $0$ .

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

Step 2.2

Set the numerator equal to zero.

$1 = 0$

Step 2.3

Since $1 \neq 0$ , 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 $\frac{1}{{(x - 2)}^{2}}$ equal to $0$ to find where the expression is undefined.

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

Step 3.2

Solve for $x$ .

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Step 3.2.1

Set the $x - 2$ equal to $0$ .

$x - 2 = 0$

Step 3.2.2

Add $2$ to both sides of the equation.

$x = 2$

Step 4

Evaluate $\frac{1}{x - 2} - 3$ at each $x$ value where the derivative is $0$ or undefined.

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Step 4.1

Evaluate at $x = 2$ .

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Step 4.1.1

Substitute $2$ for $x$ .

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

Step 4.1.2

Simplify.

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Step 4.1.2.1

Subtract $2$ from $2$ .

$\frac{1}{0} - 3$

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