Calculus Examples

$f (x) = x {(1 - 2 x)}^{3}$

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

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

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 $g (x) = 1 - 2 x$ .

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Step 2.1

To apply the Chain Rule, set $u$ as $1 - 2 x$ .

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

Step 2.2

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

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

Step 2.3

Replace all occurrences of $u$ with $1 - 2 x$ .

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

Step 3

Differentiate.

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Step 3.1

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

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

Step 3.2

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

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

Step 3.3

Add $0$ and $\frac{d}{d x} [- 2 x]$ .

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

Step 3.4

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

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

Step 3.5

Multiply $- 2$ by $3$ .

$x (- 6 {(1 - 2 x)}^{2} \frac{d}{d x} [x]) + {(1 - 2 x)}^{3} \frac{d}{d x} [x]$

Step 3.6

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

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

Step 3.7

Multiply $- 6$ by $1$ .

$x (- 6 {(1 - 2 x)}^{2}) + {(1 - 2 x)}^{3} \frac{d}{d x} [x]$

Step 3.8

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

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

Step 3.9

Multiply ${(1 - 2 x)}^{3}$ by $1$ .

$x (- 6 {(1 - 2 x)}^{2}) + {(1 - 2 x)}^{3}$

Step 4

Simplify.

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Step 4.1

Factor ${(1 - 2 x)}^{2}$ out of $x (- 6 {(1 - 2 x)}^{2}) + {(1 - 2 x)}^{3}$ .

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Step 4.1.1

Factor ${(1 - 2 x)}^{2}$ out of $x (- 6 {(1 - 2 x)}^{2})$ .

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

Step 4.1.2

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

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

Step 4.1.3

Factor ${(1 - 2 x)}^{2}$ out of ${(1 - 2 x)}^{2} (x \cdot (- 6)) + {(1 - 2 x)}^{2} (1 - 2 x)$ .

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

Step 4.2

Combine terms.

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Step 4.2.1

Move $- 6$ to the left of $x$ .

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

Step 4.2.2

Subtract $2 x$ from $- 6 x$ .

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

Step 4.3

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

$(1 - 2 x) (1 - 2 x) (- 8 x + 1)$

Step 4.4

Expand $(1 - 2 x) (1 - 2 x)$ using the FOIL Method.

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Step 4.4.1

Apply the distributive property.

$(1 (1 - 2 x) - 2 x (1 - 2 x)) (- 8 x + 1)$

Step 4.4.2

Apply the distributive property.

$(1 \cdot 1 + 1 (- 2 x) - 2 x (1 - 2 x)) (- 8 x + 1)$

Step 4.4.3

Apply the distributive property.

$(1 \cdot 1 + 1 (- 2 x) - 2 x \cdot 1 - 2 x (- 2 x)) (- 8 x + 1)$

Step 4.5

Simplify and combine like terms.

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Step 4.5.1

Simplify each term.

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Step 4.5.1.1

Multiply $1$ by $1$ .

$(1 + 1 (- 2 x) - 2 x \cdot 1 - 2 x (- 2 x)) (- 8 x + 1)$

Step 4.5.1.2

Multiply $- 2 x$ by $1$ .

$(1 - 2 x - 2 x \cdot 1 - 2 x (- 2 x)) (- 8 x + 1)$

Step 4.5.1.3

Multiply $- 2$ by $1$ .

$(1 - 2 x - 2 x - 2 x (- 2 x)) (- 8 x + 1)$

Step 4.5.1.4

Rewrite using the commutative property of multiplication.

$(1 - 2 x - 2 x - 2 \cdot - 2 x \cdot x) (- 8 x + 1)$

Step 4.5.1.5

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

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Step 4.5.1.5.1

Move $x$ .

$(1 - 2 x - 2 x - 2 \cdot - 2 (x \cdot x)) (- 8 x + 1)$

Step 4.5.1.5.2

Multiply $x$ by $x$ .

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

Step 4.5.1.6

Multiply $- 2$ by $- 2$ .

$(1 - 2 x - 2 x + 4 x^{2}) (- 8 x + 1)$

Step 4.5.2

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

$(1 - 4 x + 4 x^{2}) (- 8 x + 1)$

Step 4.6

Expand $(1 - 4 x + 4 x^{2}) (- 8 x + 1)$ by multiplying each term in the first expression by each term in the second expression.

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

Step 4.7

Simplify each term.

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Step 4.7.1

Multiply $- 8 x$ by $1$ .

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

Step 4.7.2

Multiply $1$ by $1$ .

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

Step 4.7.3

Rewrite using the commutative property of multiplication.

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

Step 4.7.4

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

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Step 4.7.4.1

Move $x$ .

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

Step 4.7.4.2

Multiply $x$ by $x$ .

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

Step 4.7.5

Multiply $- 4$ by $- 8$ .

$- 8 x + 1 + 32 x^{2} - 4 x \cdot 1 + 4 x^{2} (- 8 x) + 4 x^{2} \cdot 1$

Step 4.7.6

Multiply $- 4$ by $1$ .

$- 8 x + 1 + 32 x^{2} - 4 x + 4 x^{2} (- 8 x) + 4 x^{2} \cdot 1$

Step 4.7.7

Rewrite using the commutative property of multiplication.

$- 8 x + 1 + 32 x^{2} - 4 x + 4 \cdot - 8 x^{2} x + 4 x^{2} \cdot 1$

Step 4.7.8

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

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Step 4.7.8.1

Move $x$ .

$- 8 x + 1 + 32 x^{2} - 4 x + 4 \cdot - 8 (x \cdot x^{2}) + 4 x^{2} \cdot 1$

Step 4.7.8.2

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

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Step 4.7.8.2.1

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

$- 8 x + 1 + 32 x^{2} - 4 x + 4 \cdot - 8 (x^{1} x^{2}) + 4 x^{2} \cdot 1$

Step 4.7.8.2.2

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

$- 8 x + 1 + 32 x^{2} - 4 x + 4 \cdot - 8 x^{1 + 2} + 4 x^{2} \cdot 1$

Step 4.7.8.3

Add $1$ and $2$ .

$- 8 x + 1 + 32 x^{2} - 4 x + 4 \cdot - 8 x^{3} + 4 x^{2} \cdot 1$

Step 4.7.9

Multiply $4$ by $- 8$ .

$- 8 x + 1 + 32 x^{2} - 4 x - 32 x^{3} + 4 x^{2} \cdot 1$

Step 4.7.10

Multiply $4$ by $1$ .

$- 8 x + 1 + 32 x^{2} - 4 x - 32 x^{3} + 4 x^{2}$

Step 4.8

Subtract $4 x$ from $- 8 x$ .

$- 12 x + 1 + 32 x^{2} - 32 x^{3} + 4 x^{2}$

Step 4.9

Add $32 x^{2}$ and $4 x^{2}$ .

$- 12 x + 1 - 32 x^{3} + 36 x^{2}$