Calculus Examples

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

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 $f (x) = x^{3} - 2 x^{2} + x - 1$ and $g (x) = x - 2$ .

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

Step 2

Differentiate.

Tap for more steps...

Step 2.1

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

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

Step 2.2

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 3$ .

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

Step 2.3

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

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

Step 2.4

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

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

Step 2.5

Multiply $2$ by $- 2$ .

$\frac{(x - 2) (3 x^{2} - 4 x + \frac{d}{d x} [x] + \frac{d}{d x} [- 1]) - (x^{3} - 2 x^{2} + x - 1) \frac{d}{d x} [x - 2]}{{(x - 2)}^{2}}$

Step 2.6

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

$\frac{(x - 2) (3 x^{2} - 4 x + 1 + \frac{d}{d x} [- 1]) - (x^{3} - 2 x^{2} + x - 1) \frac{d}{d x} [x - 2]}{{(x - 2)}^{2}}$

Step 2.7

Since $- 1$ is constant with respect to $x$ , the derivative of $- 1$ with respect to $x$ is $0$ .

$\frac{(x - 2) (3 x^{2} - 4 x + 1 + 0) - (x^{3} - 2 x^{2} + x - 1) \frac{d}{d x} [x - 2]}{{(x - 2)}^{2}}$

Step 2.8

Add $3 x^{2} - 4 x + 1$ and $0$ .

$\frac{(x - 2) (3 x^{2} - 4 x + 1) - (x^{3} - 2 x^{2} + x - 1) \frac{d}{d x} [x - 2]}{{(x - 2)}^{2}}$

Step 2.9

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

$\frac{(x - 2) (3 x^{2} - 4 x + 1) - (x^{3} - 2 x^{2} + x - 1) (\frac{d}{d x} [x] + \frac{d}{d x} [- 2])}{{(x - 2)}^{2}}$

Step 2.10

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

$\frac{(x - 2) (3 x^{2} - 4 x + 1) - (x^{3} - 2 x^{2} + x - 1) (1 + \frac{d}{d x} [- 2])}{{(x - 2)}^{2}}$

Step 2.11

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

$\frac{(x - 2) (3 x^{2} - 4 x + 1) - (x^{3} - 2 x^{2} + x - 1) (1 + 0)}{{(x - 2)}^{2}}$

Step 2.12

Simplify the expression.

Tap for more steps...

Step 2.12.1

Add $1$ and $0$ .

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

Step 2.12.2

Multiply $- 1$ by $1$ .

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

Step 3

Simplify.

Tap for more steps...

Step 3.1

Apply the distributive property.

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

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 $(x - 2) (3 x^{2} - 4 x + 1)$ by multiplying each term in the first expression by each term in the second expression.

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

Step 3.2.1.2

Simplify each term.

Tap for more steps...

Step 3.2.1.2.1

Rewrite using the commutative property of multiplication.

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

Step 3.2.1.2.2

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

Tap for more steps...

Step 3.2.1.2.2.1

Move $x^{2}$ .

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

Step 3.2.1.2.2.2

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

Tap for more steps...

Step 3.2.1.2.2.2.1

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

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

Step 3.2.1.2.2.2.2

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

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

Step 3.2.1.2.2.3

Add $2$ and $1$ .

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

Step 3.2.1.2.3

Rewrite using the commutative property of multiplication.

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

Step 3.2.1.2.4

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

Tap for more steps...

Step 3.2.1.2.4.1

Move $x$ .

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

Step 3.2.1.2.4.2

Multiply $x$ by $x$ .

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

Step 3.2.1.2.5

Multiply $x$ by $1$ .

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

Step 3.2.1.2.6

Multiply $3$ by $- 2$ .

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

Step 3.2.1.2.7

Multiply $- 4$ by $- 2$ .

$\frac{3 x^{3} - 4 x^{2} + x - 6 x^{2} + 8 x - 2 \cdot 1 - x^{3} - (- 2 x^{2}) - x - - 1}{{(x - 2)}^{2}}$

Step 3.2.1.2.8

Multiply $- 2$ by $1$ .

$\frac{3 x^{3} - 4 x^{2} + x - 6 x^{2} + 8 x - 2 - x^{3} - (- 2 x^{2}) - x - - 1}{{(x - 2)}^{2}}$

Step 3.2.1.3

Subtract $6 x^{2}$ from $- 4 x^{2}$ .

$\frac{3 x^{3} - 10 x^{2} + x + 8 x - 2 - x^{3} - (- 2 x^{2}) - x - - 1}{{(x - 2)}^{2}}$

Step 3.2.1.4

Add $x$ and $8 x$ .

$\frac{3 x^{3} - 10 x^{2} + 9 x - 2 - x^{3} - (- 2 x^{2}) - x - - 1}{{(x - 2)}^{2}}$

Step 3.2.1.5

Multiply $- 2$ by $- 1$ .

$\frac{3 x^{3} - 10 x^{2} + 9 x - 2 - x^{3} + 2 x^{2} - x - - 1}{{(x - 2)}^{2}}$

Step 3.2.1.6

Multiply $- 1$ by $- 1$ .

$\frac{3 x^{3} - 10 x^{2} + 9 x - 2 - x^{3} + 2 x^{2} - x + 1}{{(x - 2)}^{2}}$

Step 3.2.2

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

$\frac{2 x^{3} - 10 x^{2} + 9 x - 2 + 2 x^{2} - x + 1}{{(x - 2)}^{2}}$

Step 3.2.3

Add $- 10 x^{2}$ and $2 x^{2}$ .

$\frac{2 x^{3} - 8 x^{2} + 9 x - 2 - x + 1}{{(x - 2)}^{2}}$

Step 3.2.4

Subtract $x$ from $9 x$ .

$\frac{2 x^{3} - 8 x^{2} + 8 x - 2 + 1}{{(x - 2)}^{2}}$

Step 3.2.5

Add $- 2$ and $1$ .

$\frac{2 x^{3} - 8 x^{2} + 8 x - 1}{{(x - 2)}^{2}}$