Calculus Examples

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

Step 1

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 $f (x) = 4 x + 1$ and $g (x) = {(1 - x)}^{3}$ .

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

Step 2

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 $g (x) = 1 - x$ .

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Step 2.1

To apply the Chain Rule, set $u$ as $1 - x$ .

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

Step 2.2

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

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

Step 2.3

Replace all occurrences of $u$ with $1 - x$ .

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

Step 3

Differentiate.

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Step 3.1

Move $3$ to the left of $4 x + 1$ .

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

Step 3.2

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

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

Step 3.3

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

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

Step 3.4

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

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

Step 3.5

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

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

Step 3.6

Multiply $- 1$ by $3$ .

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

Step 3.7

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

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

Step 3.8

Multiply $- 3$ by $1$ .

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

Step 3.9

By the Sum Rule, the derivative of $4 x + 1$ with respect to $x$ is $\frac{d}{d x} [4 x] + \frac{d}{d x} [1]$ .

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

Step 3.10

Since $4$ is constant with respect to $x$ , the derivative of $4 x$ with respect to $x$ is $4 \frac{d}{d x} [x]$ .

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

Step 3.11

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

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

Step 3.12

Multiply $4$ by $1$ .

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

Step 3.13

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

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

Step 3.14

Simplify the expression.

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Step 3.14.1

Add $4$ and $0$ .

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

Step 3.14.2

Move $4$ to the left of ${(1 - x)}^{3}$ .

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

Step 4

Simplify.

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Step 4.1

Apply the distributive property.

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

Step 4.2

Multiply $4$ by $- 3$ .

$(- 12 x - 3 \cdot 1) {(1 - x)}^{2} + 4 {(1 - x)}^{3}$

Step 4.3

Multiply $- 3$ by $1$ .

$(- 12 x - 3) {(1 - x)}^{2} + 4 {(1 - x)}^{3}$

Step 4.4

Factor ${(1 - x)}^{2}$ out of $(- 12 x - 3) {(1 - x)}^{2} + 4 {(1 - x)}^{3}$ .

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Step 4.4.1

Factor ${(1 - x)}^{2}$ out of $(- 12 x - 3) {(1 - x)}^{2}$ .

${(1 - x)}^{2} (- 12 x - 3) + 4 {(1 - x)}^{3}$

Step 4.4.2

Factor ${(1 - x)}^{2}$ out of $4 {(1 - x)}^{3}$ .

${(1 - x)}^{2} (- 12 x - 3) + {(1 - x)}^{2} (4 (1 - x))$

Step 4.4.3

Factor ${(1 - x)}^{2}$ out of ${(1 - x)}^{2} (- 12 x - 3) + {(1 - x)}^{2} (4 (1 - x))$ .

${(1 - x)}^{2} (- 12 x - 3 + 4 (1 - x))$

Step 4.5

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

$(1 - x) (1 - x) (- 12 x - 3 + 4 (1 - x))$

Step 4.6

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

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Step 4.6.1

Apply the distributive property.

$(1 (1 - x) - x (1 - x)) (- 12 x - 3 + 4 (1 - x))$

Step 4.6.2

Apply the distributive property.

$(1 \cdot 1 + 1 (- x) - x (1 - x)) (- 12 x - 3 + 4 (1 - x))$

Step 4.6.3

Apply the distributive property.

$(1 \cdot 1 + 1 (- x) - x \cdot 1 - x (- x)) (- 12 x - 3 + 4 (1 - x))$

Step 4.7

Simplify and combine like terms.

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Step 4.7.1

Simplify each term.

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Step 4.7.1.1

Multiply $1$ by $1$ .

$(1 + 1 (- x) - x \cdot 1 - x (- x)) (- 12 x - 3 + 4 (1 - x))$

Step 4.7.1.2

Multiply $- x$ by $1$ .

$(1 - x - x \cdot 1 - x (- x)) (- 12 x - 3 + 4 (1 - x))$

Step 4.7.1.3

Multiply $- 1$ by $1$ .

$(1 - x - x - x (- x)) (- 12 x - 3 + 4 (1 - x))$

Step 4.7.1.4

Rewrite using the commutative property of multiplication.

$(1 - x - x - 1 \cdot - 1 x \cdot x) (- 12 x - 3 + 4 (1 - x))$

Step 4.7.1.5

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

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Step 4.7.1.5.1

Move $x$ .

$(1 - x - x - 1 \cdot - 1 (x \cdot x)) (- 12 x - 3 + 4 (1 - x))$

Step 4.7.1.5.2

Multiply $x$ by $x$ .

$(1 - x - x - 1 \cdot - 1 x^{2}) (- 12 x - 3 + 4 (1 - x))$

Step 4.7.1.6

Multiply $- 1$ by $- 1$ .

$(1 - x - x + 1 x^{2}) (- 12 x - 3 + 4 (1 - x))$

Step 4.7.1.7

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

$(1 - x - x + x^{2}) (- 12 x - 3 + 4 (1 - x))$

Step 4.7.2

Subtract $x$ from $- x$ .

$(1 - 2 x + x^{2}) (- 12 x - 3 + 4 (1 - x))$

Step 4.8

Simplify each term.

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Step 4.8.1

Apply the distributive property.

$(1 - 2 x + x^{2}) (- 12 x - 3 + 4 \cdot 1 + 4 (- x))$

Step 4.8.2

Multiply $4$ by $1$ .

$(1 - 2 x + x^{2}) (- 12 x - 3 + 4 + 4 (- x))$

Step 4.8.3

Multiply $- 1$ by $4$ .

$(1 - 2 x + x^{2}) (- 12 x - 3 + 4 - 4 x)$

Step 4.9

Subtract $4 x$ from $- 12 x$ .

$(1 - 2 x + x^{2}) (- 16 x - 3 + 4)$

Step 4.10

Add $- 3$ and $4$ .

$(1 - 2 x + x^{2}) (- 16 x + 1)$

Step 4.11

Expand $(1 - 2 x + x^{2}) (- 16 x + 1)$ by multiplying each term in the first expression by each term in the second expression.

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

Step 4.12

Simplify each term.

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Step 4.12.1

Multiply $- 16 x$ by $1$ .

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

Step 4.12.2

Multiply $1$ by $1$ .

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

Step 4.12.3

Rewrite using the commutative property of multiplication.

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

Step 4.12.4

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

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Step 4.12.4.1

Move $x$ .

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

Step 4.12.4.2

Multiply $x$ by $x$ .

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

Step 4.12.5

Multiply $- 2$ by $- 16$ .

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

Step 4.12.6

Multiply $- 2$ by $1$ .

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

Step 4.12.7

Rewrite using the commutative property of multiplication.

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

Step 4.12.8

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

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Step 4.12.8.1

Move $x$ .

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

Step 4.12.8.2

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

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Step 4.12.8.2.1

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

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

Step 4.12.8.2.2

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

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

Step 4.12.8.3

Add $1$ and $2$ .

$- 16 x + 1 + 32 x^{2} - 2 x - 16 x^{3} + x^{2} \cdot 1$

Step 4.12.9

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

$- 16 x + 1 + 32 x^{2} - 2 x - 16 x^{3} + x^{2}$

Step 4.13

Subtract $2 x$ from $- 16 x$ .

$- 18 x + 1 + 32 x^{2} - 16 x^{3} + x^{2}$

Step 4.14

Add $32 x^{2}$ and $x^{2}$ .

$- 18 x + 1 - 16 x^{3} + 33 x^{2}$