Calculus Examples

$f (x) = x {(x - 4)}^{3} + 3$

Step 1

By the Sum Rule, the derivative of $x {(x - 4)}^{3} + 3$ with respect to $x$ is $\frac{d}{d x} [x {(x - 4)}^{3}] + \frac{d}{d x} [3]$ .

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

Step 2

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

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Step 2.1

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

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

Step 2.2

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 $g (x) = x - 4$ .

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Step 2.2.1

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

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

Step 2.2.2

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

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

Step 2.2.3

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

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

Step 2.3

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

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

Step 2.4

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

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

Step 2.5

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

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

Step 2.6

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

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

Step 2.7

Add $1$ and $0$ .

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

Step 2.8

Multiply $3$ by $1$ .

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

Step 2.9

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

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

Step 3

Since $3$ is constant with respect to $x$ , the derivative of $3$ with respect to $x$ is $0$ .

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

Step 4

Simplify.

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Step 4.1

Add $x (3 {(x - 4)}^{2}) + {(x - 4)}^{3}$ and $0$ .

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

Step 4.2

Reorder terms.

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

Step 4.3

Simplify each term.

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Step 4.3.1

Use the Binomial Theorem.

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

Step 4.3.2

Simplify each term.

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Step 4.3.2.1

Multiply $- 4$ by $3$ .

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

Step 4.3.2.2

Raise $- 4$ to the power of $2$ .

$x^{3} - 12 x^{2} + 3 x \cdot 16 + {(- 4)}^{3} + 3 x {(x - 4)}^{2}$

Step 4.3.2.3

Multiply $16$ by $3$ .

$x^{3} - 12 x^{2} + 48 x + {(- 4)}^{3} + 3 x {(x - 4)}^{2}$

Step 4.3.2.4

Raise $- 4$ to the power of $3$ .

$x^{3} - 12 x^{2} + 48 x - 64 + 3 x {(x - 4)}^{2}$

Step 4.3.3

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

$x^{3} - 12 x^{2} + 48 x - 64 + 3 x ((x - 4) (x - 4))$

Step 4.3.4

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

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Step 4.3.4.1

Apply the distributive property.

$x^{3} - 12 x^{2} + 48 x - 64 + 3 x (x (x - 4) - 4 (x - 4))$

Step 4.3.4.2

Apply the distributive property.

$x^{3} - 12 x^{2} + 48 x - 64 + 3 x (x \cdot x + x \cdot - 4 - 4 (x - 4))$

Step 4.3.4.3

Apply the distributive property.

$x^{3} - 12 x^{2} + 48 x - 64 + 3 x (x \cdot x + x \cdot - 4 - 4 x - 4 \cdot - 4)$

Step 4.3.5

Simplify and combine like terms.

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Step 4.3.5.1

Simplify each term.

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Step 4.3.5.1.1

Multiply $x$ by $x$ .

$x^{3} - 12 x^{2} + 48 x - 64 + 3 x (x^{2} + x \cdot - 4 - 4 x - 4 \cdot - 4)$

Step 4.3.5.1.2

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

$x^{3} - 12 x^{2} + 48 x - 64 + 3 x (x^{2} - 4 \cdot x - 4 x - 4 \cdot - 4)$

Step 4.3.5.1.3

Multiply $- 4$ by $- 4$ .

$x^{3} - 12 x^{2} + 48 x - 64 + 3 x (x^{2} - 4 x - 4 x + 16)$

Step 4.3.5.2

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

$x^{3} - 12 x^{2} + 48 x - 64 + 3 x (x^{2} - 8 x + 16)$

Step 4.3.6

Apply the distributive property.

$x^{3} - 12 x^{2} + 48 x - 64 + 3 x \cdot x^{2} + 3 x (- 8 x) + 3 x \cdot 16$

Step 4.3.7

Simplify.

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Step 4.3.7.1

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

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Step 4.3.7.1.1

Move $x^{2}$ .

$x^{3} - 12 x^{2} + 48 x - 64 + 3 (x^{2} x) + 3 x (- 8 x) + 3 x \cdot 16$

Step 4.3.7.1.2

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

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Step 4.3.7.1.2.1

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

$x^{3} - 12 x^{2} + 48 x - 64 + 3 (x^{2} x^{1}) + 3 x (- 8 x) + 3 x \cdot 16$

Step 4.3.7.1.2.2

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

$x^{3} - 12 x^{2} + 48 x - 64 + 3 x^{2 + 1} + 3 x (- 8 x) + 3 x \cdot 16$

Step 4.3.7.1.3

Add $2$ and $1$ .

$x^{3} - 12 x^{2} + 48 x - 64 + 3 x^{3} + 3 x (- 8 x) + 3 x \cdot 16$

Step 4.3.7.2

Rewrite using the commutative property of multiplication.

$x^{3} - 12 x^{2} + 48 x - 64 + 3 x^{3} + 3 \cdot - 8 x \cdot x + 3 x \cdot 16$

Step 4.3.7.3

Multiply $16$ by $3$ .

$x^{3} - 12 x^{2} + 48 x - 64 + 3 x^{3} + 3 \cdot - 8 x \cdot x + 48 x$

Step 4.3.8

Simplify each term.

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Step 4.3.8.1

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

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Step 4.3.8.1.1

Move $x$ .

$x^{3} - 12 x^{2} + 48 x - 64 + 3 x^{3} + 3 \cdot - 8 (x \cdot x) + 48 x$

Step 4.3.8.1.2

Multiply $x$ by $x$ .

$x^{3} - 12 x^{2} + 48 x - 64 + 3 x^{3} + 3 \cdot - 8 x^{2} + 48 x$

Step 4.3.8.2

Multiply $3$ by $- 8$ .

$x^{3} - 12 x^{2} + 48 x - 64 + 3 x^{3} - 24 x^{2} + 48 x$

Step 4.4

Add $x^{3}$ and $3 x^{3}$ .

$4 x^{3} - 12 x^{2} + 48 x - 64 - 24 x^{2} + 48 x$

Step 4.5

Subtract $24 x^{2}$ from $- 12 x^{2}$ .

$4 x^{3} - 36 x^{2} + 48 x - 64 + 48 x$

Step 4.6

Add $48 x$ and $48 x$ .

$4 x^{3} - 36 x^{2} + 96 x - 64$