Calculus Examples

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

Step 1

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

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

Step 2

Expand $(x - 1) (x - 1)$ using the FOIL Method.

Tap for more steps...

Step 2.1

Apply the distributive property.

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

Step 2.2

Apply the distributive property.

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

Step 2.3

Apply the distributive property.

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

Step 3

Simplify and combine like terms.

Tap for more steps...

Step 3.1

Simplify each term.

Tap for more steps...

Step 3.1.1

Multiply $x$ by $x$ .

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

Step 3.1.2

Move $- 1$ to the left of $x$ .

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

Step 3.1.3

Rewrite $- 1 x$ as $- x$ .

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

Step 3.1.4

Rewrite $- 1 x$ as $- x$ .

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

Step 3.1.5

Multiply $- 1$ by $- 1$ .

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

Step 3.2

Subtract $x$ from $- x$ .

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

Step 4

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

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

Step 5

Differentiate using the Product Rule which states that $\frac{d}{d x} [f (x) g (x)]$ is $f (x) \frac{d}{d x} [g (x)] + g (x) \frac{d}{d x} [f (x)]$ where $f (x) = x + 1$ and $g (x) = x^{2} - 2 x + 1$ .

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

Step 6

Differentiate.

Tap for more steps...

Step 6.1

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

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

Step 6.2

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

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

Step 6.3

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

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

Step 6.4

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

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

Step 6.5

Multiply $- 2$ by $1$ .

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

Step 6.6

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

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

Step 6.7

Add $2 x - 2$ and $0$ .

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

Step 6.8

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

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

Step 6.9

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

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

Step 6.10

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

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

Step 6.11

Simplify the expression.

Tap for more steps...

Step 6.11.1

Add $1$ and $0$ .

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

Step 6.11.2

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

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

Step 7

Simplify.

Tap for more steps...

Step 7.1

Apply the distributive property.

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

Step 7.2

Apply the distributive property.

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

Step 7.3

Apply the distributive property.

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

Step 7.4

Apply the distributive property.

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

Step 7.5

Apply the distributive property.

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

Step 7.6

Combine terms.

Tap for more steps...

Step 7.6.1

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

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

Step 7.6.2

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

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

Step 7.6.3

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

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

Step 7.6.4

Add $1$ and $1$ .

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

Step 7.6.5

Multiply $2$ by $- 1$ .

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

Step 7.6.6

Multiply $2$ by $1$ .

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

Step 7.6.7

Multiply $2$ by $- 1$ .

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

Step 7.6.8

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

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

Step 7.6.9

Multiply $- 2$ by $- 1$ .

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

Step 7.6.10

Multiply $- 2$ by $1$ .

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

Step 7.6.11

Multiply $- 1$ by $- 2$ .

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

Step 7.6.12

Add $- 2 x$ and $2 x$ .

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

Step 7.6.13

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

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

Step 7.6.14

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

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

Step 7.6.15

Multiply $- 2$ by $- 1$ .

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

Step 7.6.16

Multiply $- 1$ by $1$ .

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

Step 7.6.17

Subtract $1$ from $2$ .

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