Calculus Examples

$y = \frac{3}{x - 2}$ , $[4, 7]$

Step 1

Find the derivative.

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Step 1.1

Find the first derivative.

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Step 1.1.1

Differentiate using the Constant Multiple Rule.

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Step 1.1.1.1

Since $3$ is constant with respect to $x$ , the derivative of $\frac{3}{x - 2}$ with respect to $x$ is $3 \frac{d}{d x} [\frac{1}{x - 2}]$ .

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

Step 1.1.1.2

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

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

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 $g (x) = x - 2$ .

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Step 1.1.2.1

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

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

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 $n = - 1$ .

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

Step 1.1.2.3

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

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

Step 1.1.3

Differentiate.

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Step 1.1.3.1

Multiply $- 1$ by $3$ .

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

Step 1.1.3.2

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

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

Step 1.1.3.3

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

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

Step 1.1.3.4

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

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

Step 1.1.3.5

Simplify the expression.

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Step 1.1.3.5.1

Add $1$ and $0$ .

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

Step 1.1.3.5.2

Multiply $- 3$ by $1$ .

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

Step 1.1.4

Simplify.

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Step 1.1.4.1

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

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

Step 1.1.4.2

Combine terms.

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Step 1.1.4.2.1

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

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

Step 1.1.4.2.2

Move the negative in front of the fraction.

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

Step 1.2

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

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

Step 2

Find if the derivative is continuous on $[4, 7]$ .

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Step 2.1

To find whether the function is continuous on $[4, 7]$ or not, find the domain of $f' (x) = - \frac{3}{{(x - 2)}^{2}}$ .

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Step 2.1.1

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

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

Step 2.1.2

Solve for $x$ .

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Step 2.1.2.1

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

$x - 2 = 0$

Step 2.1.2.2

Add $2$ to both sides of the equation.

$x = 2$

Step 2.1.3

The domain is all values of $x$ that make the expression defined.

Interval Notation:

$(- \infty, 2) \cup (2, \infty)$

Set-Builder Notation:

${x | x \neq 2}$

Interval Notation:

$(- \infty, 2) \cup (2, \infty)$

Set-Builder Notation:

${x | x \neq 2}$

Step 2.2

$f' (x)$ is continuous on $[4, 7]$ .

The function is continuous.

Step 3

The function is differentiable on $[4, 7]$ because the derivative is continuous on $[4, 7]$ .

The function is differentiable.

Step 4