Calculus Examples

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

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

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

Step 2

Differentiate.

Tap for more steps...

Step 2.1

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

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

Step 2.2

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

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

Step 2.3

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

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

Step 2.4

Simplify the expression.

Tap for more steps...

Step 2.4.1

Add $2 x$ and $0$ .

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

Step 2.4.2

Move $2$ to the left of $x^{2} - 2 x$ .

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

Step 2.5

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

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

Step 2.6

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

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

Step 2.7

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

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

Step 2.8

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

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

Step 2.9

Multiply $- 2$ by $1$ .

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

Step 3

Simplify.

Tap for more steps...

Step 3.1

Apply the distributive property.

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

Step 3.2

Apply the distributive property.

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

Step 3.3

Apply the distributive property.

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

Step 3.4

Simplify the numerator.

Tap for more steps...

Step 3.4.1

Simplify each term.

Tap for more steps...

Step 3.4.1.1

Multiply $x^{2}$ by $x$ by adding the exponents.

Tap for more steps...

Step 3.4.1.1.1

Move $x$ .

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

Step 3.4.1.1.2

Multiply $x$ by $x^{2}$ .

Tap for more steps...

Step 3.4.1.1.2.1

Raise $x$ to the power of $1$ .

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

Step 3.4.1.1.2.2

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

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

Step 3.4.1.1.3

Add $1$ and $2$ .

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

Step 3.4.1.2

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

Tap for more steps...

Step 3.4.1.2.1

Move $x$ .

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

Step 3.4.1.2.2

Multiply $x$ by $x$ .

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

Step 3.4.1.3

Multiply $- 2$ by $2$ .

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

Step 3.4.1.4

Multiply $- 1$ by $- 4$ .

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

Step 3.4.1.5

Expand $(- x^{2} + 4) (2 x - 2)$ using the FOIL Method.

Tap for more steps...

Step 3.4.1.5.1

Apply the distributive property.

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

Step 3.4.1.5.2

Apply the distributive property.

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

Step 3.4.1.5.3

Apply the distributive property.

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

Step 3.4.1.6

Simplify each term.

Tap for more steps...

Step 3.4.1.6.1

Rewrite using the commutative property of multiplication.

$\frac{2 x^{3} - 4 x^{2} - 1 \cdot 2 x^{2} x - x^{2} \cdot - 2 + 4 (2 x) + 4 \cdot - 2}{{(x^{2} - 2 x)}^{2}}$

Step 3.4.1.6.2

Multiply $x^{2}$ by $x$ by adding the exponents.

Tap for more steps...

Step 3.4.1.6.2.1

Move $x$ .

$\frac{2 x^{3} - 4 x^{2} - 1 \cdot 2 (x \cdot x^{2}) - x^{2} \cdot - 2 + 4 (2 x) + 4 \cdot - 2}{{(x^{2} - 2 x)}^{2}}$

Step 3.4.1.6.2.2

Multiply $x$ by $x^{2}$ .

Tap for more steps...

Step 3.4.1.6.2.2.1

Raise $x$ to the power of $1$ .

$\frac{2 x^{3} - 4 x^{2} - 1 \cdot 2 (x^{1} x^{2}) - x^{2} \cdot - 2 + 4 (2 x) + 4 \cdot - 2}{{(x^{2} - 2 x)}^{2}}$

Step 3.4.1.6.2.2.2

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$\frac{2 x^{3} - 4 x^{2} - 1 \cdot 2 x^{1 + 2} - x^{2} \cdot - 2 + 4 (2 x) + 4 \cdot - 2}{{(x^{2} - 2 x)}^{2}}$

Step 3.4.1.6.2.3

Add $1$ and $2$ .

$\frac{2 x^{3} - 4 x^{2} - 1 \cdot 2 x^{3} - x^{2} \cdot - 2 + 4 (2 x) + 4 \cdot - 2}{{(x^{2} - 2 x)}^{2}}$

Step 3.4.1.6.3

Multiply $- 1$ by $2$ .

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

Step 3.4.1.6.4

Multiply $- 2$ by $- 1$ .

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

Step 3.4.1.6.5

Multiply $2$ by $4$ .

$\frac{2 x^{3} - 4 x^{2} - 2 x^{3} + 2 x^{2} + 8 x + 4 \cdot - 2}{{(x^{2} - 2 x)}^{2}}$

Step 3.4.1.6.6

Multiply $4$ by $- 2$ .

$\frac{2 x^{3} - 4 x^{2} - 2 x^{3} + 2 x^{2} + 8 x - 8}{{(x^{2} - 2 x)}^{2}}$

Step 3.4.2

Combine the opposite terms in $2 x^{3} - 4 x^{2} - 2 x^{3} + 2 x^{2} + 8 x - 8$ .

Tap for more steps...

Step 3.4.2.1

Subtract $2 x^{3}$ from $2 x^{3}$ .

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

Step 3.4.2.2

Add $- 4 x^{2}$ and $0$ .

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

Step 3.4.3

Add $- 4 x^{2}$ and $2 x^{2}$ .

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

Step 3.5

Simplify the numerator.

Tap for more steps...

Step 3.5.1

Factor $2$ out of $- 2 x^{2} + 8 x - 8$ .

Tap for more steps...

Step 3.5.1.1

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

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

Step 3.5.1.2

Factor $2$ out of $8 x$ .

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

Step 3.5.1.3

Factor $2$ out of $- 8$ .

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

Step 3.5.1.4

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

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

Step 3.5.1.5

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

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

Step 3.5.2

Factor by grouping.

Tap for more steps...

Step 3.5.2.1

For a polynomial of the form $a x^{2} + b x + c$ , rewrite the middle term as a sum of two terms whose product is $a \cdot c = - 1 \cdot - 4 = 4$ and whose sum is $b = 4$ .

Tap for more steps...

Step 3.5.2.1.1

Factor $4$ out of $4 x$ .

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

Step 3.5.2.1.2

Rewrite $4$ as $2$ plus $2$

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

Step 3.5.2.1.3

Apply the distributive property.

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

Step 3.5.2.2

Factor out the greatest common factor from each group.

Tap for more steps...

Step 3.5.2.2.1

Group the first two terms and the last two terms.

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

Step 3.5.2.2.2

Factor out the greatest common factor (GCF) from each group.

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

Step 3.5.2.3

Factor the polynomial by factoring out the greatest common factor, $- x + 2$ .

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

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

Step 3.5.3

Combine exponents.

Tap for more steps...

Step 3.5.3.1

Factor $- 1$ out of $- x$ .

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

Step 3.5.3.2

Rewrite $2$ as $- 1 (- 2)$ .

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

Step 3.5.3.3

Factor $- 1$ out of $- (x) - 1 (- 2)$ .

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

Step 3.5.3.4

Rewrite $- (x - 2)$ as $- 1 (x - 2)$ .

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

Step 3.5.3.5

Raise $x - 2$ to the power of $1$ .

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

Step 3.5.3.6

Raise $x - 2$ to the power of $1$ .

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

Step 3.5.3.7

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

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

Step 3.5.3.8

Add $1$ and $1$ .

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

Step 3.5.3.9

Multiply $2$ by $- 1$ .

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

Step 3.6

Simplify the denominator.

Tap for more steps...

Step 3.6.1

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

Tap for more steps...

Step 3.6.1.1

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

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

Step 3.6.1.2

Factor $x$ out of $- 2 x$ .

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

Step 3.6.1.3

Factor $x$ out of $x \cdot x + x \cdot - 2$ .

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

Step 3.6.2

Apply the product rule to $x (x - 2)$ .

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

Step 3.7

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

Tap for more steps...

Step 3.7.1

Cancel the common factor.

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

Step 3.7.2

Rewrite the expression.

$\frac{- 2}{x^{2}}$

Step 3.8

Move the negative in front of the fraction.

$- \frac{2}{x^{2}}$