Calculus Examples

$y = \frac{4 x^{2} - 16}{x - 2}$

Step 1

Find the first derivative.

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Step 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 $f (x) = 4 x^{2} - 16$ and $g (x) = x - 2$ .

$\frac{(x - 2) \frac{d}{d x} [4 x^{2} - 16] - (4 x^{2} - 16) \frac{d}{d x} [x - 2]}{{(x - 2)}^{2}}$

Step 1.2

Differentiate.

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Step 1.2.1

By the Sum Rule, the derivative of $4 x^{2} - 16$ with respect to $x$ is $\frac{d}{d x} [4 x^{2}] + \frac{d}{d x} [- 16]$ .

$\frac{(x - 2) (\frac{d}{d x} [4 x^{2}] + \frac{d}{d x} [- 16]) - (4 x^{2} - 16) \frac{d}{d x} [x - 2]}{{(x - 2)}^{2}}$

Step 1.2.2

Since $4$ is constant with respect to $x$ , the derivative of $4 x^{2}$ with respect to $x$ is $4 \frac{d}{d x} [x^{2}]$ .

$\frac{(x - 2) (4 \frac{d}{d x} [x^{2}] + \frac{d}{d x} [- 16]) - (4 x^{2} - 16) \frac{d}{d x} [x - 2]}{{(x - 2)}^{2}}$

Step 1.2.3

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 2$ .

$\frac{(x - 2) (4 (2 x) + \frac{d}{d x} [- 16]) - (4 x^{2} - 16) \frac{d}{d x} [x - 2]}{{(x - 2)}^{2}}$

Step 1.2.4

Multiply $2$ by $4$ .

$\frac{(x - 2) (8 x + \frac{d}{d x} [- 16]) - (4 x^{2} - 16) \frac{d}{d x} [x - 2]}{{(x - 2)}^{2}}$

Step 1.2.5

Since $- 16$ is constant with respect to $x$ , the derivative of $- 16$ with respect to $x$ is $0$ .

$\frac{(x - 2) (8 x + 0) - (4 x^{2} - 16) \frac{d}{d x} [x - 2]}{{(x - 2)}^{2}}$

Step 1.2.6

Simplify the expression.

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Step 1.2.6.1

Add $8 x$ and $0$ .

$\frac{(x - 2) (8 x) - (4 x^{2} - 16) \frac{d}{d x} [x - 2]}{{(x - 2)}^{2}}$

Step 1.2.6.2

Move $8$ to the left of $x - 2$ .

$\frac{8 \cdot (x - 2) x - (4 x^{2} - 16) \frac{d}{d x} [x - 2]}{{(x - 2)}^{2}}$

Step 1.2.7

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

$\frac{8 (x - 2) x - (4 x^{2} - 16) (\frac{d}{d x} [x] + \frac{d}{d x} [- 2])}{{(x - 2)}^{2}}$

Step 1.2.8

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

$\frac{8 (x - 2) x - (4 x^{2} - 16) (1 + \frac{d}{d x} [- 2])}{{(x - 2)}^{2}}$

Step 1.2.9

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

$\frac{8 (x - 2) x - (4 x^{2} - 16) (1 + 0)}{{(x - 2)}^{2}}$

Step 1.2.10

Simplify the expression.

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Step 1.2.10.1

Add $1$ and $0$ .

$\frac{8 (x - 2) x - (4 x^{2} - 16) \cdot 1}{{(x - 2)}^{2}}$

Step 1.2.10.2

Multiply $- 1$ by $1$ .

$\frac{8 (x - 2) x - (4 x^{2} - 16)}{{(x - 2)}^{2}}$

Step 1.3

Simplify.

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Step 1.3.1

Apply the distributive property.

$\frac{(8 x + 8 \cdot - 2) x - (4 x^{2} - 16)}{{(x - 2)}^{2}}$

Step 1.3.2

Apply the distributive property.

$\frac{8 x \cdot x + 8 \cdot - 2 x - (4 x^{2} - 16)}{{(x - 2)}^{2}}$

Step 1.3.3

Apply the distributive property.

$\frac{8 x \cdot x + 8 \cdot - 2 x - (4 x^{2}) - - 16}{{(x - 2)}^{2}}$

Step 1.3.4

Simplify the numerator.

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Step 1.3.4.1

Simplify each term.

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Step 1.3.4.1.1

Multiply $x$ by $x$ by adding the exponents.

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Step 1.3.4.1.1.1

Move $x$ .

$\frac{8 (x \cdot x) + 8 \cdot - 2 x - (4 x^{2}) - - 16}{{(x - 2)}^{2}}$

Step 1.3.4.1.1.2

Multiply $x$ by $x$ .

$\frac{8 x^{2} + 8 \cdot - 2 x - (4 x^{2}) - - 16}{{(x - 2)}^{2}}$

Step 1.3.4.1.2

Multiply $8$ by $- 2$ .

$\frac{8 x^{2} - 16 x - (4 x^{2}) - - 16}{{(x - 2)}^{2}}$

Step 1.3.4.1.3

Multiply $4$ by $- 1$ .

$\frac{8 x^{2} - 16 x - 4 x^{2} - - 16}{{(x - 2)}^{2}}$

Step 1.3.4.1.4

Multiply $- 1$ by $- 16$ .

$\frac{8 x^{2} - 16 x - 4 x^{2} + 16}{{(x - 2)}^{2}}$

Step 1.3.4.2

Subtract $4 x^{2}$ from $8 x^{2}$ .

$\frac{4 x^{2} - 16 x + 16}{{(x - 2)}^{2}}$

Step 1.3.5

Simplify the numerator.

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Step 1.3.5.1

Factor $4$ out of $4 x^{2} - 16 x + 16$ .

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Step 1.3.5.1.1

Factor $4$ out of $4 x^{2}$ .

$\frac{4 (x^{2}) - 16 x + 16}{{(x - 2)}^{2}}$

Step 1.3.5.1.2

Factor $4$ out of $- 16 x$ .

$\frac{4 (x^{2}) + 4 (- 4 x) + 16}{{(x - 2)}^{2}}$

Step 1.3.5.1.3

Factor $4$ out of $16$ .

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

Step 1.3.5.1.4

Factor $4$ out of $4 x^{2} + 4 (- 4 x)$ .

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

Step 1.3.5.1.5

Factor $4$ out of $4 (x^{2} - 4 x) + 4 \cdot 4$ .

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

Step 1.3.5.2

Factor using the perfect square rule.

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Step 1.3.5.2.1

Rewrite $4$ as $2^{2}$ .

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

Step 1.3.5.2.2

Check that the middle term is two times the product of the numbers being squared in the first term and third term.

$4 x = 2 \cdot x \cdot 2$

Step 1.3.5.2.3

Rewrite the polynomial.

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

Step 1.3.5.2.4

Factor using the perfect square trinomial rule $a^{2} - 2 a b + b^{2} = {(a - b)}^{2}$ , where $a = x$ and $b = 2$ .

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

Step 1.3.6

Cancel the common factor of ${(x - 2)}^{2}$ .

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Step 1.3.6.1

Cancel the common factor.

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

Step 1.3.6.2

Divide $4$ by $1$ .

$f' (x) = 4$

Step 2

Since $4$ is constant with respect to $x$ , the derivative of $4$ with respect to $x$ is $0$ .

$f'' (x) = 0$