Calculus Examples

$2 x {(x - 4)}^{3}$

Step 1

Since $2$ is constant with respect to $x$ , the derivative of $2 x {(x - 4)}^{3}$ with respect to $x$ is $2 \frac{d}{d x} [x {(x - 4)}^{3}]$ .

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

Step 2

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 $f (x) = x$ and $g (x) = {(x - 4)}^{3}$ .

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

Step 3

Differentiate using the chain rule, which states that $\frac{d}{d x} [f (g (x))]$ is $f' (g (x)) g' (x)$ where $f (x) = x^{3}$ and $g (x) = x - 4$ .

Tap for more steps...

Step 3.1

To apply the Chain Rule, set $u$ as $x - 4$ .

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

Step 3.2

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

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

Step 3.3

Replace all occurrences of $u$ with $x - 4$ .

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

Step 4

Differentiate.

Tap for more steps...

Step 4.1

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

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

Step 4.2

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

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

Step 4.3

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

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

Step 4.4

Simplify the expression.

Tap for more steps...

Step 4.4.1

Add $1$ and $0$ .

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

Step 4.4.2

Multiply $3$ by $1$ .

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

Step 4.5

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

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

Step 4.6

Multiply ${(x - 4)}^{3}$ by $1$ .

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

Step 5

Simplify.

Tap for more steps...

Step 5.1

Apply the distributive property.

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

Step 5.2

Multiply $3$ by $2$ .

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

Step 5.3

Factor $2 {(x - 4)}^{2}$ out of $6 x {(x - 4)}^{2} + 2 {(x - 4)}^{3}$ .

Tap for more steps...

Step 5.3.1

Factor $2 {(x - 4)}^{2}$ out of $6 x {(x - 4)}^{2}$ .

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

Step 5.3.2

Factor $2 {(x - 4)}^{2}$ out of $2 {(x - 4)}^{3}$ .

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

Step 5.3.3

Factor $2 {(x - 4)}^{2}$ out of $2 {(x - 4)}^{2} (3 x) + 2 {(x - 4)}^{2} (x - 4)$ .

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

Step 5.4

Add $3 x$ and $x$ .

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

Step 5.5

Rewrite ${(x - 4)}^{2}$ as $(x - 4) (x - 4)$ .

$2 ((x - 4) (x - 4)) (4 x - 4)$

Step 5.6

Expand $(x - 4) (x - 4)$ using the FOIL Method.

Tap for more steps...

Step 5.6.1

Apply the distributive property.

$2 (x (x - 4) - 4 (x - 4)) (4 x - 4)$

Step 5.6.2

Apply the distributive property.

$2 (x \cdot x + x \cdot - 4 - 4 (x - 4)) (4 x - 4)$

Step 5.6.3

Apply the distributive property.

$2 (x \cdot x + x \cdot - 4 - 4 x - 4 \cdot - 4) (4 x - 4)$

Step 5.7

Simplify and combine like terms.

Tap for more steps...

Step 5.7.1

Simplify each term.

Tap for more steps...

Step 5.7.1.1

Multiply $x$ by $x$ .

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

Step 5.7.1.2

Move $- 4$ to the left of $x$ .

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

Step 5.7.1.3

Multiply $- 4$ by $- 4$ .

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

Step 5.7.2

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

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

Step 5.8

Apply the distributive property.

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

Step 5.9

Simplify.

Tap for more steps...

Step 5.9.1

Multiply $- 8$ by $2$ .

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

Step 5.9.2

Multiply $2$ by $16$ .

$(2 x^{2} - 16 x + 32) (4 x - 4)$

Step 5.10

Expand $(2 x^{2} - 16 x + 32) (4 x - 4)$ by multiplying each term in the first expression by each term in the second expression.

$2 x^{2} (4 x) + 2 x^{2} \cdot - 4 - 16 x (4 x) - 16 x \cdot - 4 + 32 (4 x) + 32 \cdot - 4$

Step 5.11

Simplify each term.

Tap for more steps...

Step 5.11.1

Rewrite using the commutative property of multiplication.

$2 \cdot 4 x^{2} x + 2 x^{2} \cdot - 4 - 16 x (4 x) - 16 x \cdot - 4 + 32 (4 x) + 32 \cdot - 4$

Step 5.11.2

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

Tap for more steps...

Step 5.11.2.1

Move $x$ .

$2 \cdot 4 (x \cdot x^{2}) + 2 x^{2} \cdot - 4 - 16 x (4 x) - 16 x \cdot - 4 + 32 (4 x) + 32 \cdot - 4$

Step 5.11.2.2

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

Tap for more steps...

Step 5.11.2.2.1

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

$2 \cdot 4 (x^{1} x^{2}) + 2 x^{2} \cdot - 4 - 16 x (4 x) - 16 x \cdot - 4 + 32 (4 x) + 32 \cdot - 4$

Step 5.11.2.2.2

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

$2 \cdot 4 x^{1 + 2} + 2 x^{2} \cdot - 4 - 16 x (4 x) - 16 x \cdot - 4 + 32 (4 x) + 32 \cdot - 4$

Step 5.11.2.3

Add $1$ and $2$ .

$2 \cdot 4 x^{3} + 2 x^{2} \cdot - 4 - 16 x (4 x) - 16 x \cdot - 4 + 32 (4 x) + 32 \cdot - 4$

Step 5.11.3

Multiply $2$ by $4$ .

$8 x^{3} + 2 x^{2} \cdot - 4 - 16 x (4 x) - 16 x \cdot - 4 + 32 (4 x) + 32 \cdot - 4$

Step 5.11.4

Multiply $- 4$ by $2$ .

$8 x^{3} - 8 x^{2} - 16 x (4 x) - 16 x \cdot - 4 + 32 (4 x) + 32 \cdot - 4$

Step 5.11.5

Rewrite using the commutative property of multiplication.

$8 x^{3} - 8 x^{2} - 16 \cdot 4 x \cdot x - 16 x \cdot - 4 + 32 (4 x) + 32 \cdot - 4$

Step 5.11.6

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

Tap for more steps...

Step 5.11.6.1

Move $x$ .

$8 x^{3} - 8 x^{2} - 16 \cdot 4 (x \cdot x) - 16 x \cdot - 4 + 32 (4 x) + 32 \cdot - 4$

Step 5.11.6.2

Multiply $x$ by $x$ .

$8 x^{3} - 8 x^{2} - 16 \cdot 4 x^{2} - 16 x \cdot - 4 + 32 (4 x) + 32 \cdot - 4$

Step 5.11.7

Multiply $- 16$ by $4$ .

$8 x^{3} - 8 x^{2} - 64 x^{2} - 16 x \cdot - 4 + 32 (4 x) + 32 \cdot - 4$

Step 5.11.8

Multiply $- 4$ by $- 16$ .

$8 x^{3} - 8 x^{2} - 64 x^{2} + 64 x + 32 (4 x) + 32 \cdot - 4$

Step 5.11.9

Multiply $4$ by $32$ .

$8 x^{3} - 8 x^{2} - 64 x^{2} + 64 x + 128 x + 32 \cdot - 4$

Step 5.11.10

Multiply $32$ by $- 4$ .

$8 x^{3} - 8 x^{2} - 64 x^{2} + 64 x + 128 x - 128$

Step 5.12

Subtract $64 x^{2}$ from $- 8 x^{2}$ .

$8 x^{3} - 72 x^{2} + 64 x + 128 x - 128$

Step 5.13

Add $64 x$ and $128 x$ .

$8 x^{3} - 72 x^{2} + 192 x - 128$