Calculus Examples

$y = x {(x - 4)}^{3}$

Step 1

Find the first derivative.

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Step 1.1

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

$f' (x) = x \frac{d}{d x} ({(x - 4)}^{3}) + {(x - 4)}^{3} \frac{d}{d x} (x)$

Step 1.2

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 1.2.1

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

$f' (x) = x (\frac{d}{d u} (u^{3}) \frac{d}{d x} (x - 4)) + {(x - 4)}^{3} \frac{d}{d x} (x)$

Step 1.2.2

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

$f' (x) = x (3 u^{2} \frac{d}{d x} (x - 4)) + {(x - 4)}^{3} \frac{d}{d x} (x)$

Step 1.2.3

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

$f' (x) = x (3 {(x - 4)}^{2} \frac{d}{d x} (x - 4)) + {(x - 4)}^{3} \frac{d}{d x} (x)$

Step 1.3

Differentiate.

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Step 1.3.1

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

$f' (x) = x (3 {(x - 4)}^{2} (\frac{d}{d x} (x) + \frac{d}{d x} (- 4))) + {(x - 4)}^{3} \frac{d}{d x} (x)$

Step 1.3.2

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

$f' (x) = x (3 {(x - 4)}^{2} (1 + \frac{d}{d x} (- 4))) + {(x - 4)}^{3} \frac{d}{d x} (x)$

Step 1.3.3

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

$f' (x) = x (3 {(x - 4)}^{2} (1 + 0)) + {(x - 4)}^{3} \frac{d}{d x} (x)$

Step 1.3.4

Simplify the expression.

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Step 1.3.4.1

Add $1$ and $0$ .

$f' (x) = x (3 {(x - 4)}^{2} \cdot 1) + {(x - 4)}^{3} \frac{d}{d x} (x)$

Step 1.3.4.2

Multiply $3$ by $1$ .

$f' (x) = x (3 {(x - 4)}^{2}) + {(x - 4)}^{3} \frac{d}{d x} (x)$

Step 1.3.5

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

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

Step 1.3.6

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

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

Step 1.4

Simplify.

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Step 1.4.1

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

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Step 1.4.1.1

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

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

Step 1.4.1.2

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

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

Step 1.4.1.3

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

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

Step 1.4.2

Combine terms.

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Step 1.4.2.1

Move $3$ to the left of $x$ .

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

Step 1.4.2.2

Add $3 x$ and $x$ .

$f' (x) = {(x - 4)}^{2} (4 x - 4)$

Step 1.4.3

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

$f' (x) = (x - 4) (x - 4) (4 x - 4)$

Step 1.4.4

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

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Step 1.4.4.1

Apply the distributive property.

$f' (x) = (x (x - 4) - 4 (x - 4)) (4 x - 4)$

Step 1.4.4.2

Apply the distributive property.

$f' (x) = (x \cdot x + x \cdot - 4 - 4 (x - 4)) (4 x - 4)$

Step 1.4.4.3

Apply the distributive property.

$f' (x) = (x \cdot x + x \cdot - 4 - 4 x - 4 \cdot - 4) (4 x - 4)$

Step 1.4.5

Simplify and combine like terms.

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Step 1.4.5.1

Simplify each term.

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Step 1.4.5.1.1

Multiply $x$ by $x$ .

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

Step 1.4.5.1.2

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

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

Step 1.4.5.1.3

Multiply $- 4$ by $- 4$ .

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

Step 1.4.5.2

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

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

Step 1.4.6

Expand $(x^{2} - 8 x + 16) (4 x - 4)$ by multiplying each term in the first expression by each term in the second expression.

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

Step 1.4.7

Simplify each term.

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Step 1.4.7.1

Rewrite using the commutative property of multiplication.

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

Step 1.4.7.2

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

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Step 1.4.7.2.1

Move $x$ .

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

Step 1.4.7.2.2

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

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Step 1.4.7.2.2.1

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

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

Step 1.4.7.2.2.2

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

$f' (x) = 4 x^{1 + 2} + x^{2} \cdot - 4 - 8 x (4 x) - 8 x \cdot - 4 + 16 (4 x) + 16 \cdot - 4$

Step 1.4.7.2.3

Add $1$ and $2$ .

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

Step 1.4.7.3

Move $- 4$ to the left of $x^{2}$ .

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

Step 1.4.7.4

Rewrite using the commutative property of multiplication.

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

Step 1.4.7.5

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

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Step 1.4.7.5.1

Move $x$ .

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

Step 1.4.7.5.2

Multiply $x$ by $x$ .

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

Step 1.4.7.6

Multiply $- 8$ by $4$ .

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

Step 1.4.7.7

Multiply $- 4$ by $- 8$ .

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

Step 1.4.7.8

Multiply $4$ by $16$ .

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

Step 1.4.7.9

Multiply $16$ by $- 4$ .

$f' (x) = 4 x^{3} - 4 x^{2} - 32 x^{2} + 32 x + 64 x - 64$

Step 1.4.8

Subtract $32 x^{2}$ from $- 4 x^{2}$ .

$f' (x) = 4 x^{3} - 36 x^{2} + 32 x + 64 x - 64$

Step 1.4.9

Add $32 x$ and $64 x$ .

$f' (x) = 4 x^{3} - 36 x^{2} + 96 x - 64$

Step 2

Find the second derivative.

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Step 2.1

By the Sum Rule, the derivative of $4 x^{3} - 36 x^{2} + 96 x - 64$ with respect to $x$ is $\frac{d}{d x} [4 x^{3}] + \frac{d}{d x} [- 36 x^{2}] + \frac{d}{d x} [96 x] + \frac{d}{d x} [- 64]$ .

$f'' (x) = \frac{d}{d x} (4 x^{3}) + \frac{d}{d x} (- 36 x^{2}) + \frac{d}{d x} (96 x) + \frac{d}{d x} (- 64)$

Step 2.2

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

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Step 2.2.1

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

$f'' (x) = 4 \frac{d}{d x} (x^{3}) + \frac{d}{d x} (- 36 x^{2}) + \frac{d}{d x} (96 x) + \frac{d}{d x} (- 64)$

Step 2.2.2

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

$f'' (x) = 4 (3 x^{2}) + \frac{d}{d x} (- 36 x^{2}) + \frac{d}{d x} (96 x) + \frac{d}{d x} (- 64)$

Step 2.2.3

Multiply $3$ by $4$ .

$f'' (x) = 12 x^{2} + \frac{d}{d x} (- 36 x^{2}) + \frac{d}{d x} (96 x) + \frac{d}{d x} (- 64)$

Step 2.3

Evaluate $\frac{d}{d x} [- 36 x^{2}]$ .

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Step 2.3.1

Since $- 36$ is constant with respect to $x$ , the derivative of $- 36 x^{2}$ with respect to $x$ is $- 36 \frac{d}{d x} [x^{2}]$ .

$f'' (x) = 12 x^{2} - 36 \frac{d}{d x} x^{2} + \frac{d}{d x} (96 x) + \frac{d}{d x} (- 64)$

Step 2.3.2

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 2$ .

$f'' (x) = 12 x^{2} - 36 (2 x) + \frac{d}{d x} (96 x) + \frac{d}{d x} (- 64)$

Step 2.3.3

Multiply $2$ by $- 36$ .

$f'' (x) = 12 x^{2} - 72 x + \frac{d}{d x} (96 x) + \frac{d}{d x} (- 64)$

Step 2.4

Evaluate $\frac{d}{d x} [96 x]$ .

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Step 2.4.1

Since $96$ is constant with respect to $x$ , the derivative of $96 x$ with respect to $x$ is $96 \frac{d}{d x} [x]$ .

$f'' (x) = 12 x^{2} - 72 x + 96 \frac{d}{d x} (x) + \frac{d}{d x} (- 64)$

Step 2.4.2

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

$f'' (x) = 12 x^{2} - 72 x + 96 \cdot 1 + \frac{d}{d x} (- 64)$

Step 2.4.3

Multiply $96$ by $1$ .

$f'' (x) = 12 x^{2} - 72 x + 96 + \frac{d}{d x} (- 64)$

Step 2.5

Differentiate using the Constant Rule.

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Step 2.5.1

Since $- 64$ is constant with respect to $x$ , the derivative of $- 64$ with respect to $x$ is $0$ .

$f'' (x) = 12 x^{2} - 72 x + 96 + 0$

Step 2.5.2

Add $12 x^{2} - 72 x + 96$ and $0$ .

$f'' (x) = 12 x^{2} - 72 x + 96$

Step 3

Find the third derivative.

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Step 3.1

By the Sum Rule, the derivative of $12 x^{2} - 72 x + 96$ with respect to $x$ is $\frac{d}{d x} [12 x^{2}] + \frac{d}{d x} [- 72 x] + \frac{d}{d x} [96]$ .

$f''' (x) = \frac{d}{d x} (12 x^{2}) + \frac{d}{d x} (- 72 x) + \frac{d}{d x} (96)$

Step 3.2

Evaluate $\frac{d}{d x} [12 x^{2}]$ .

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Step 3.2.1

Since $12$ is constant with respect to $x$ , the derivative of $12 x^{2}$ with respect to $x$ is $12 \frac{d}{d x} [x^{2}]$ .

$f''' (x) = 12 \frac{d}{d x} (x^{2}) + \frac{d}{d x} (- 72 x) + \frac{d}{d x} (96)$

Step 3.2.2

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 2$ .

$f''' (x) = 12 (2 x) + \frac{d}{d x} (- 72 x) + \frac{d}{d x} (96)$

Step 3.2.3

Multiply $2$ by $12$ .

$f''' (x) = 24 x + \frac{d}{d x} (- 72 x) + \frac{d}{d x} (96)$

Step 3.3

Evaluate $\frac{d}{d x} [- 72 x]$ .

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Step 3.3.1

Since $- 72$ is constant with respect to $x$ , the derivative of $- 72 x$ with respect to $x$ is $- 72 \frac{d}{d x} [x]$ .

$f''' (x) = 24 x - 72 \frac{d}{d x} x + \frac{d}{d x} (96)$

Step 3.3.2

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

$f''' (x) = 24 x - 72 \cdot 1 + \frac{d}{d x} (96)$

Step 3.3.3

Multiply $- 72$ by $1$ .

$f''' (x) = 24 x - 72 + \frac{d}{d x} (96)$

Step 3.4

Differentiate using the Constant Rule.

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Step 3.4.1

Since $96$ is constant with respect to $x$ , the derivative of $96$ with respect to $x$ is $0$ .

$f''' (x) = 24 x - 72 + 0$

Step 3.4.2

Add $24 x - 72$ and $0$ .

$f''' (x) = 24 x - 72$

Step 4

Find the fourth derivative.

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Step 4.1

By the Sum Rule, the derivative of $24 x - 72$ with respect to $x$ is $\frac{d}{d x} [24 x] + \frac{d}{d x} [- 72]$ .

$f^{4} (x) = \frac{d}{d x} (24 x) + \frac{d}{d x} (- 72)$

Step 4.2

Evaluate $\frac{d}{d x} [24 x]$ .

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Step 4.2.1

Since $24$ is constant with respect to $x$ , the derivative of $24 x$ with respect to $x$ is $24 \frac{d}{d x} [x]$ .

$f^{4} (x) = 24 \frac{d}{d x} (x) + \frac{d}{d x} (- 72)$

Step 4.2.2

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

$f^{4} (x) = 24 \cdot 1 + \frac{d}{d x} (- 72)$

Step 4.2.3

Multiply $24$ by $1$ .

$f^{4} (x) = 24 + \frac{d}{d x} (- 72)$

Step 4.3

Differentiate using the Constant Rule.

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Step 4.3.1

Since $- 72$ is constant with respect to $x$ , the derivative of $- 72$ with respect to $x$ is $0$ .

$f^{4} (x) = 24 + 0$

Step 4.3.2

Add $24$ and $0$ .

$f^{4} (x) = 24$