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

$x^{2}$ .

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