Calculus Examples

x {(x - 4)}^{3}

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

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

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

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

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

$x - 4$ .

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

Step 1.2.2

Differentiate using the Power Rule which states that

$\frac{d}{d u} [u^{n}]$ is

$n u^{n - 1}$ where

$n = 3$ .

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

Step 1.2.3

Replace all occurrences of

$u$ with

$x - 4$ .

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

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

$\frac{d}{d x} [x] + \frac{d}{d x} [- 4]$ .

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

Step 1.3.2

Differentiate using the Power Rule which states that

$\frac{d}{d x} [x^{n}]$ is

$n x^{n - 1}$ where

$n = 1$ .

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

Step 1.3.3

Since

$- 4$ is constant with respect to

$x$ , the derivative of

$- 4$ with respect to

$x$ is

$0$ .

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

Step 1.3.4

Simplify the expression.

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Step 1.3.4.1

Add

$1$ and

$0$ .

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

Step 1.3.4.2

Multiply

$3$ by

$1$ .

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

Step 1.3.5

Differentiate using the Power Rule which states that

$\frac{d}{d x} [x^{n}]$ is

$n x^{n - 1}$ where

$n = 1$ .

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

Step 1.3.6

Multiply

${(x - 4)}^{3}$ by

$1$ .

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

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

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

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

Step 1.4.1.2

Factor

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

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

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

Step 1.4.1.3

Factor

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

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

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

Step 1.4.2

Combine terms.

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Step 1.4.2.1

Move

$3$ to the left of

$x$ .

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

Step 1.4.2.2

Add

$3 x$ and

$x$ .

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

Step 1.4.3

Rewrite

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

$(x - 4) (x - 4)$ .

$(x - 4) (x - 4) (4 x - 4)$

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.

$(x (x - 4) - 4 (x - 4)) (4 x - 4)$

Step 1.4.4.2

Apply the distributive property.

$(x \cdot x + x \cdot - 4 - 4 (x - 4)) (4 x - 4)$

Step 1.4.4.3

Apply the distributive property.

$(x \cdot x + x \cdot - 4 - 4 x - 4 \cdot - 4) (4 x - 4)$

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

$x$ .

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

Step 1.4.5.1.2

Move

$- 4$ to the left of

$x$ .

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

Step 1.4.5.1.3

Multiply

$- 4$ by

$- 4$ .

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

Step 1.4.5.2

Subtract

$4 x$ from

$- 4 x$ .

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

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.

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

Step 1.4.7

Simplify each term.

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Step 1.4.7.1

Rewrite using the commutative property of multiplication.

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

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

$x$ .

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

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

$1$ .

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

Step 1.4.7.2.2.2

Use the power rule

$a^{m} a^{n} = a^{m + n}$ to combine exponents.

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

Step 1.4.7.2.3

Add

$1$ and

$2$ .

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

Step 1.4.7.3

Move

$- 4$ to the left of

$x^{2}$ .

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

Step 1.4.7.4

Rewrite using the commutative property of multiplication.

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

Step 1.4.7.5

Multiply

$x$ by

$x$ by adding the exponents.

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Step 1.4.7.5.1

Move

$x$ .

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

Step 1.4.7.5.2

Multiply

$x$ by

$x$ .

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

Step 1.4.7.6

Multiply

$- 8$ by

$4$ .

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

Step 1.4.7.7

Multiply

$- 4$ by

$- 8$ .

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

Step 1.4.7.8

Multiply

$4$ by

$16$ .

$4 x^{3} - 4 x^{2} - 32 x^{2} + 32 x + 64 x + 16 \cdot - 4$

Step 1.4.7.9

Multiply

$16$ by

$- 4$ .

$4 x^{3} - 4 x^{2} - 32 x^{2} + 32 x + 64 x - 64$

Step 1.4.8

Subtract

$32 x^{2}$ from

$- 4 x^{2}$ .

$4 x^{3} - 36 x^{2} + 32 x + 64 x - 64$

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

$\frac{d}{d x} [4 x^{3}] + \frac{d}{d x} [- 36 x^{2}] + \frac{d}{d x} [96 x] + \frac{d}{d x} [- 64]$ .

$\frac{d}{d x} [4 x^{3}] + \frac{d}{d x} [- 36 x^{2}] + \frac{d}{d x} [96 x] + \frac{d}{d x} [- 64]$

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

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

$4 \frac{d}{d x} [x^{3}] + \frac{d}{d x} [- 36 x^{2}] + \frac{d}{d x} [96 x] + \frac{d}{d x} [- 64]$

Step 2.2.2

Differentiate using the Power Rule which states that

$\frac{d}{d x} [x^{n}]$ is

$n x^{n - 1}$ where

$n = 3$ .

$4 (3 x^{2}) + \frac{d}{d x} [- 36 x^{2}] + \frac{d}{d x} [96 x] + \frac{d}{d x} [- 64]$

Step 2.2.3

Multiply

$3$ by

$4$ .

$12 x^{2} + \frac{d}{d x} [- 36 x^{2}] + \frac{d}{d x} [96 x] + \frac{d}{d x} [- 64]$

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

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

$12 x^{2} - 36 \frac{d}{d x} [x^{2}] + \frac{d}{d x} [96 x] + \frac{d}{d x} [- 64]$

Step 2.3.2

Differentiate using the Power Rule which states that

$\frac{d}{d x} [x^{n}]$ is

$n x^{n - 1}$ where

$n = 2$ .

$12 x^{2} - 36 (2 x) + \frac{d}{d x} [96 x] + \frac{d}{d x} [- 64]$

Step 2.3.3

Multiply

$2$ by

$- 36$ .

$12 x^{2} - 72 x + \frac{d}{d x} [96 x] + \frac{d}{d x} [- 64]$

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

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

$12 x^{2} - 72 x + 96 \frac{d}{d x} [x] + \frac{d}{d x} [- 64]$

Step 2.4.2

Differentiate using the Power Rule which states that

$\frac{d}{d x} [x^{n}]$ is

$n x^{n - 1}$ where

$n = 1$ .

$12 x^{2} - 72 x + 96 \cdot 1 + \frac{d}{d x} [- 64]$

Step 2.4.3

Multiply

$96$ by

$1$ .

$12 x^{2} - 72 x + 96 + \frac{d}{d x} [- 64]$

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

$0$ .

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

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

$\frac{d}{d x} [12 x^{2}] + \frac{d}{d x} [- 72 x] + \frac{d}{d x} [96]$ .

$\frac{d}{d x} [12 x^{2}] + \frac{d}{d x} [- 72 x] + \frac{d}{d x} [96]$

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

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

$12 \frac{d}{d x} [x^{2}] + \frac{d}{d x} [- 72 x] + \frac{d}{d x} [96]$

Step 3.2.2

Differentiate using the Power Rule which states that

$\frac{d}{d x} [x^{n}]$ is

$n x^{n - 1}$ where

$n = 2$ .

$12 (2 x) + \frac{d}{d x} [- 72 x] + \frac{d}{d x} [96]$

Step 3.2.3

Multiply

$2$ by

$12$ .

$24 x + \frac{d}{d x} [- 72 x] + \frac{d}{d x} [96]$

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

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

$24 x - 72 \frac{d}{d x} [x] + \frac{d}{d x} [96]$

Step 3.3.2

Differentiate using the Power Rule which states that

$\frac{d}{d x} [x^{n}]$ is

$n x^{n - 1}$ where

$n = 1$ .

$24 x - 72 \cdot 1 + \frac{d}{d x} [96]$

Step 3.3.3

Multiply

$- 72$ by

$1$ .

$24 x - 72 + \frac{d}{d x} [96]$

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

$0$ .

$24 x - 72 + 0$

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

$\frac{d}{d x} [24 x] + \frac{d}{d x} [- 72]$ .

$\frac{d}{d x} [24 x] + \frac{d}{d x} [- 72]$

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

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

$24 \frac{d}{d x} [x] + \frac{d}{d x} [- 72]$

Step 4.2.2

Differentiate using the Power Rule which states that

$\frac{d}{d x} [x^{n}]$ is

$n x^{n - 1}$ where

$n = 1$ .

$24 \cdot 1 + \frac{d}{d x} [- 72]$

Step 4.2.3

Multiply

$24$ by

$1$ .

$24 + \frac{d}{d x} [- 72]$

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

$0$ .

$24 + 0$

Step 4.3.2

Add

$24$ and

$0$ .

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