Calculus Examples

$\frac{x - x^{2}}{2 - 3 x + x^{2}}$

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 $f (x) = x - x^{2}$ and $g (x) = 2 - 3 x + x^{2}$ .

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

Step 2

Differentiate.

Tap for more steps...

Step 2.1

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

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

Step 2.2

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

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

Step 2.3

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

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

Step 2.4

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

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

Step 2.5

Multiply $2$ by $- 1$ .

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

Step 2.6

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

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

Step 2.7

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

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

Step 2.8

Add $0$ and $\frac{d}{d x} [- 3 x]$ .

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

Step 2.9

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

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

Step 2.10

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

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

Step 2.11

Multiply $- 3$ by $1$ .

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

Step 2.12

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

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

Step 3

Simplify.

Tap for more steps...

Step 3.1

Apply the distributive property.

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

Step 3.2

Simplify the numerator.

Tap for more steps...

Step 3.2.1

Simplify each term.

Tap for more steps...

Step 3.2.1.1

Expand $(2 - 3 x + x^{2}) (1 - 2 x)$ by multiplying each term in the first expression by each term in the second expression.

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

Step 3.2.1.2

Simplify each term.

Tap for more steps...

Step 3.2.1.2.1

Multiply $2$ by $1$ .

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

Step 3.2.1.2.2

Multiply $- 2$ by $2$ .

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

Step 3.2.1.2.3

Multiply $- 3$ by $1$ .

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

Step 3.2.1.2.4

Rewrite using the commutative property of multiplication.

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

Step 3.2.1.2.5

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

Tap for more steps...

Step 3.2.1.2.5.1

Move $x$ .

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

Step 3.2.1.2.5.2

Multiply $x$ by $x$ .

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

Step 3.2.1.2.6

Multiply $- 3$ by $- 2$ .

$\frac{2 - 4 x - 3 x + 6 x^{2} + x^{2} \cdot 1 + x^{2} (- 2 x) + (- x - - x^{2}) (- 3 + 2 x)}{{(2 - 3 x + x^{2})}^{2}}$

Step 3.2.1.2.7

Multiply $x^{2}$ by $1$ .

$\frac{2 - 4 x - 3 x + 6 x^{2} + x^{2} + x^{2} (- 2 x) + (- x - - x^{2}) (- 3 + 2 x)}{{(2 - 3 x + x^{2})}^{2}}$

Step 3.2.1.2.8

Rewrite using the commutative property of multiplication.

$\frac{2 - 4 x - 3 x + 6 x^{2} + x^{2} - 2 x^{2} x + (- x - - x^{2}) (- 3 + 2 x)}{{(2 - 3 x + x^{2})}^{2}}$

Step 3.2.1.2.9

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

Tap for more steps...

Step 3.2.1.2.9.1

Move $x$ .

$\frac{2 - 4 x - 3 x + 6 x^{2} + x^{2} - 2 (x \cdot x^{2}) + (- x - - x^{2}) (- 3 + 2 x)}{{(2 - 3 x + x^{2})}^{2}}$

Step 3.2.1.2.9.2

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

Tap for more steps...

Step 3.2.1.2.9.2.1

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

$\frac{2 - 4 x - 3 x + 6 x^{2} + x^{2} - 2 (x^{1} x^{2}) + (- x - - x^{2}) (- 3 + 2 x)}{{(2 - 3 x + x^{2})}^{2}}$

Step 3.2.1.2.9.2.2

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

$\frac{2 - 4 x - 3 x + 6 x^{2} + x^{2} - 2 x^{1 + 2} + (- x - - x^{2}) (- 3 + 2 x)}{{(2 - 3 x + x^{2})}^{2}}$

Step 3.2.1.2.9.3

Add $1$ and $2$ .

$\frac{2 - 4 x - 3 x + 6 x^{2} + x^{2} - 2 x^{3} + (- x - - x^{2}) (- 3 + 2 x)}{{(2 - 3 x + x^{2})}^{2}}$

Step 3.2.1.3

Subtract $3 x$ from $- 4 x$ .

$\frac{2 - 7 x + 6 x^{2} + x^{2} - 2 x^{3} + (- x - - x^{2}) (- 3 + 2 x)}{{(2 - 3 x + x^{2})}^{2}}$

Step 3.2.1.4

Add $6 x^{2}$ and $x^{2}$ .

$\frac{2 - 7 x + 7 x^{2} - 2 x^{3} + (- x - - x^{2}) (- 3 + 2 x)}{{(2 - 3 x + x^{2})}^{2}}$

Step 3.2.1.5

Multiply $- - x^{2}$ .

Tap for more steps...

Step 3.2.1.5.1

Multiply $- 1$ by $- 1$ .

$\frac{2 - 7 x + 7 x^{2} - 2 x^{3} + (- x + 1 x^{2}) (- 3 + 2 x)}{{(2 - 3 x + x^{2})}^{2}}$

Step 3.2.1.5.2

Multiply $x^{2}$ by $1$ .

$\frac{2 - 7 x + 7 x^{2} - 2 x^{3} + (- x + x^{2}) (- 3 + 2 x)}{{(2 - 3 x + x^{2})}^{2}}$

Step 3.2.1.6

Expand $(- x + x^{2}) (- 3 + 2 x)$ using the FOIL Method.

Tap for more steps...

Step 3.2.1.6.1

Apply the distributive property.

$\frac{2 - 7 x + 7 x^{2} - 2 x^{3} - x (- 3 + 2 x) + x^{2} (- 3 + 2 x)}{{(2 - 3 x + x^{2})}^{2}}$

Step 3.2.1.6.2

Apply the distributive property.

$\frac{2 - 7 x + 7 x^{2} - 2 x^{3} - x \cdot - 3 - x (2 x) + x^{2} (- 3 + 2 x)}{{(2 - 3 x + x^{2})}^{2}}$

Step 3.2.1.6.3

Apply the distributive property.

$\frac{2 - 7 x + 7 x^{2} - 2 x^{3} - x \cdot - 3 - x (2 x) + x^{2} \cdot - 3 + x^{2} (2 x)}{{(2 - 3 x + x^{2})}^{2}}$

Step 3.2.1.7

Simplify and combine like terms.

Tap for more steps...

Step 3.2.1.7.1

Simplify each term.

Tap for more steps...

Step 3.2.1.7.1.1

Multiply $- 3$ by $- 1$ .

$\frac{2 - 7 x + 7 x^{2} - 2 x^{3} + 3 x - x (2 x) + x^{2} \cdot - 3 + x^{2} (2 x)}{{(2 - 3 x + x^{2})}^{2}}$

Step 3.2.1.7.1.2

Rewrite using the commutative property of multiplication.

$\frac{2 - 7 x + 7 x^{2} - 2 x^{3} + 3 x - 1 \cdot 2 x \cdot x + x^{2} \cdot - 3 + x^{2} (2 x)}{{(2 - 3 x + x^{2})}^{2}}$

Step 3.2.1.7.1.3

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

Tap for more steps...

Step 3.2.1.7.1.3.1

Move $x$ .

$\frac{2 - 7 x + 7 x^{2} - 2 x^{3} + 3 x - 1 \cdot 2 (x \cdot x) + x^{2} \cdot - 3 + x^{2} (2 x)}{{(2 - 3 x + x^{2})}^{2}}$

Step 3.2.1.7.1.3.2

Multiply $x$ by $x$ .

$\frac{2 - 7 x + 7 x^{2} - 2 x^{3} + 3 x - 1 \cdot 2 x^{2} + x^{2} \cdot - 3 + x^{2} (2 x)}{{(2 - 3 x + x^{2})}^{2}}$

Step 3.2.1.7.1.4

Multiply $- 1$ by $2$ .

$\frac{2 - 7 x + 7 x^{2} - 2 x^{3} + 3 x - 2 x^{2} + x^{2} \cdot - 3 + x^{2} (2 x)}{{(2 - 3 x + x^{2})}^{2}}$

Step 3.2.1.7.1.5

Move $- 3$ to the left of $x^{2}$ .

$\frac{2 - 7 x + 7 x^{2} - 2 x^{3} + 3 x - 2 x^{2} - 3 \cdot x^{2} + x^{2} (2 x)}{{(2 - 3 x + x^{2})}^{2}}$

Step 3.2.1.7.1.6

Rewrite using the commutative property of multiplication.

$\frac{2 - 7 x + 7 x^{2} - 2 x^{3} + 3 x - 2 x^{2} - 3 x^{2} + 2 x^{2} x}{{(2 - 3 x + x^{2})}^{2}}$

Step 3.2.1.7.1.7

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

Tap for more steps...

Step 3.2.1.7.1.7.1

Move $x$ .

$\frac{2 - 7 x + 7 x^{2} - 2 x^{3} + 3 x - 2 x^{2} - 3 x^{2} + 2 (x \cdot x^{2})}{{(2 - 3 x + x^{2})}^{2}}$

Step 3.2.1.7.1.7.2

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

Tap for more steps...

Step 3.2.1.7.1.7.2.1

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

$\frac{2 - 7 x + 7 x^{2} - 2 x^{3} + 3 x - 2 x^{2} - 3 x^{2} + 2 (x^{1} x^{2})}{{(2 - 3 x + x^{2})}^{2}}$

Step 3.2.1.7.1.7.2.2

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

$\frac{2 - 7 x + 7 x^{2} - 2 x^{3} + 3 x - 2 x^{2} - 3 x^{2} + 2 x^{1 + 2}}{{(2 - 3 x + x^{2})}^{2}}$

Step 3.2.1.7.1.7.3

Add $1$ and $2$ .

$\frac{2 - 7 x + 7 x^{2} - 2 x^{3} + 3 x - 2 x^{2} - 3 x^{2} + 2 x^{3}}{{(2 - 3 x + x^{2})}^{2}}$

Step 3.2.1.7.2

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

$\frac{2 - 7 x + 7 x^{2} - 2 x^{3} + 3 x - 5 x^{2} + 2 x^{3}}{{(2 - 3 x + x^{2})}^{2}}$

Step 3.2.2

Combine the opposite terms in $2 - 7 x + 7 x^{2} - 2 x^{3} + 3 x - 5 x^{2} + 2 x^{3}$ .

Tap for more steps...

Step 3.2.2.1

Add $- 2 x^{3}$ and $2 x^{3}$ .

$\frac{2 - 7 x + 7 x^{2} + 3 x - 5 x^{2} + 0}{{(2 - 3 x + x^{2})}^{2}}$

Step 3.2.2.2

Add $2 - 7 x + 7 x^{2} + 3 x - 5 x^{2}$ and $0$ .

$\frac{2 - 7 x + 7 x^{2} + 3 x - 5 x^{2}}{{(2 - 3 x + x^{2})}^{2}}$

Step 3.2.3

Add $- 7 x$ and $3 x$ .

$\frac{2 + 7 x^{2} - 4 x - 5 x^{2}}{{(2 - 3 x + x^{2})}^{2}}$

Step 3.2.4

Subtract $5 x^{2}$ from $7 x^{2}$ .

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

Step 3.3

Reorder terms.

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

Step 3.4

Simplify the numerator.

Tap for more steps...

Step 3.4.1

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

Tap for more steps...

Step 3.4.1.1

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

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

Step 3.4.1.2

Factor $2$ out of $- 4 x$ .

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

Step 3.4.1.3

Factor $2$ out of $2$ .

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

Step 3.4.1.4

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

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

Step 3.4.1.5

Factor $2$ out of $2 (x^{2} - 2 x) + 2 (1)$ .

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

Step 3.4.2

Factor using the perfect square rule.

Tap for more steps...

Step 3.4.2.1

Rewrite $1$ as $1^{2}$ .

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

Step 3.4.2.2

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

$2 x = 2 \cdot x \cdot 1$

Step 3.4.2.3

Rewrite the polynomial.

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

Step 3.4.2.4

Factor using the perfect square trinomial rule $a^{2} - 2 a b + b^{2} = {(a - b)}^{2}$ , where $a = x$ and $b = 1$ .

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

Step 3.5

Simplify the denominator.

Tap for more steps...

Step 3.5.1

Factor $x^{2} - 3 x + 2$ using the AC method.

Tap for more steps...

Step 3.5.1.1

Consider the form $x^{2} + b x + c$ . Find a pair of integers whose product is $c$ and whose sum is $b$ . In this case, whose product is $2$ and whose sum is $- 3$ .

$- 2, - 1$

Step 3.5.1.2

Write the factored form using these integers.

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

Step 3.5.2

Apply the product rule to $(x - 2) (x - 1)$ .

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

Step 3.6

Cancel the common factor of ${(x - 1)}^{2}$ .

Tap for more steps...

Step 3.6.1

Cancel the common factor.

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

Step 3.6.2

Rewrite the expression.

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