Calculus Examples

${(a + b)}^{4} - {(a - b)}^{4}$

Step 1

Find the first derivative.

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Step 1.1

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

$\frac{d}{d a} [{(a + b)}^{4}] + \frac{d}{d a} [- {(a - b)}^{4}]$

Step 1.2

Evaluate $\frac{d}{d a} [{(a + b)}^{4}]$ .

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Step 1.2.1

Differentiate using the chain rule, which states that $\frac{d}{d a} [f (g (a))]$ is $f' (g (a)) g' (a)$ where $f (a) = a^{4}$ and $g (a) = a + b$ .

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Step 1.2.1.1

To apply the Chain Rule, set $u_{1}$ as $a + b$ .

$\frac{d}{d u_{1}} [{u_{1}}^{4}] \frac{d}{d a} [a + b] + \frac{d}{d a} [- {(a - b)}^{4}]$

Step 1.2.1.2

Differentiate using the Power Rule which states that $\frac{d}{d u_{1}} [{u_{1}}^{n}]$ is $n {u_{1}}^{n - 1}$ where $n = 4$ .

$4 {u_{1}}^{3} \frac{d}{d a} [a + b] + \frac{d}{d a} [- {(a - b)}^{4}]$

Step 1.2.1.3

Replace all occurrences of $u_{1}$ with $a + b$ .

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

Step 1.2.2

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

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

Step 1.2.3

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

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

Step 1.2.4

Since $b$ is constant with respect to $a$ , the derivative of $b$ with respect to $a$ is $0$ .

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

Step 1.2.5

Add $1$ and $0$ .

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

Step 1.2.6

Multiply $4$ by $1$ .

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

Step 1.3

Evaluate $\frac{d}{d a} [- {(a - b)}^{4}]$ .

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Step 1.3.1

Since $- 1$ is constant with respect to $a$ , the derivative of $- {(a - b)}^{4}$ with respect to $a$ is $- \frac{d}{d a} [{(a - b)}^{4}]$ .

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

Step 1.3.2

Differentiate using the chain rule, which states that $\frac{d}{d a} [f (g (a))]$ is $f' (g (a)) g' (a)$ where $f (a) = a^{4}$ and $g (a) = a - b$ .

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Step 1.3.2.1

To apply the Chain Rule, set $u_{2}$ as $a - b$ .

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

Step 1.3.2.2

Differentiate using the Power Rule which states that $\frac{d}{d u_{2}} [{u_{2}}^{n}]$ is $n {u_{2}}^{n - 1}$ where $n = 4$ .

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

Step 1.3.2.3

Replace all occurrences of $u_{2}$ with $a - b$ .

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

Step 1.3.3

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

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

Step 1.3.4

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

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

Step 1.3.5

Since $- b$ is constant with respect to $a$ , the derivative of $- b$ with respect to $a$ is $0$ .

$4 {(a + b)}^{3} - (4 {(a - b)}^{3} (1 + 0))$

Step 1.3.6

Add $1$ and $0$ .

$4 {(a + b)}^{3} - (4 {(a - b)}^{3} \cdot 1)$

Step 1.3.7

Multiply $4$ by $1$ .

$4 {(a + b)}^{3} - (4 {(a - b)}^{3})$

Step 1.3.8

Multiply $4$ by $- 1$ .

$4 {(a + b)}^{3} - 4 {(a - b)}^{3}$

Step 1.4

Simplify.

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Step 1.4.1

Factor $4$ out of $4 {(a + b)}^{3} - 4 {(a - b)}^{3}$ .

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Step 1.4.1.1

Factor $4$ out of $4 {(a + b)}^{3}$ .

$4 ({(a + b)}^{3}) - 4 {(a - b)}^{3}$

Step 1.4.1.2

Factor $4$ out of $- 4 {(a - b)}^{3}$ .

$4 ({(a + b)}^{3}) + 4 (- {(a - b)}^{3})$

Step 1.4.1.3

Factor $4$ out of $4 ({(a + b)}^{3}) + 4 (- {(a - b)}^{3})$ .

$4 ({(a + b)}^{3} - {(a - b)}^{3})$

Step 1.4.2

Simplify each term.

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Step 1.4.2.1

Use the Binomial Theorem.

$4 (a^{3} + 3 a^{2} b + 3 a b^{2} + b^{3} - {(a - b)}^{3})$

Step 1.4.2.2

Use the Binomial Theorem.

$4 (a^{3} + 3 a^{2} b + 3 a b^{2} + b^{3} - (a^{3} + 3 a^{2} (- b) + 3 a {(- b)}^{2} + {(- b)}^{3}))$

Step 1.4.2.3

Simplify each term.

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Step 1.4.2.3.1

Rewrite using the commutative property of multiplication.

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

Step 1.4.2.3.2

Multiply $3$ by $- 1$ .

$4 (a^{3} + 3 a^{2} b + 3 a b^{2} + b^{3} - (a^{3} - 3 a^{2} b + 3 a {(- b)}^{2} + {(- b)}^{3}))$

Step 1.4.2.3.3

Apply the product rule to $- b$ .

$4 (a^{3} + 3 a^{2} b + 3 a b^{2} + b^{3} - (a^{3} - 3 a^{2} b + 3 a ({(- 1)}^{2} b^{2}) + {(- b)}^{3}))$

Step 1.4.2.3.4

Rewrite using the commutative property of multiplication.

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

Step 1.4.2.3.5

Raise $- 1$ to the power of $2$ .

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

Step 1.4.2.3.6

Multiply $3$ by $1$ .

$4 (a^{3} + 3 a^{2} b + 3 a b^{2} + b^{3} - (a^{3} - 3 a^{2} b + 3 a b^{2} + {(- b)}^{3}))$

Step 1.4.2.3.7

Apply the product rule to $- b$ .

$4 (a^{3} + 3 a^{2} b + 3 a b^{2} + b^{3} - (a^{3} - 3 a^{2} b + 3 a b^{2} + {(- 1)}^{3} b^{3}))$

Step 1.4.2.3.8

Raise $- 1$ to the power of $3$ .

$4 (a^{3} + 3 a^{2} b + 3 a b^{2} + b^{3} - (a^{3} - 3 a^{2} b + 3 a b^{2} - b^{3}))$

Step 1.4.2.4

Apply the distributive property.

$4 (a^{3} + 3 a^{2} b + 3 a b^{2} + b^{3} - a^{3} - (- 3 a^{2} b) - (3 a b^{2}) - - b^{3})$

Step 1.4.2.5

Simplify.

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Step 1.4.2.5.1

Multiply $- 3$ by $- 1$ .

$4 (a^{3} + 3 a^{2} b + 3 a b^{2} + b^{3} - a^{3} + 3 (a^{2} b) - (3 a b^{2}) - - b^{3})$

Step 1.4.2.5.2

Multiply $3$ by $- 1$ .

$4 (a^{3} + 3 a^{2} b + 3 a b^{2} + b^{3} - a^{3} + 3 (a^{2} b) - 3 (a b^{2}) - - b^{3})$

Step 1.4.2.5.3

Multiply $- - b^{3}$ .

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Step 1.4.2.5.3.1

Multiply $- 1$ by $- 1$ .

$4 (a^{3} + 3 a^{2} b + 3 a b^{2} + b^{3} - a^{3} + 3 (a^{2} b) - 3 (a b^{2}) + 1 b^{3})$

Step 1.4.2.5.3.2

Multiply $b^{3}$ by $1$ .

$4 (a^{3} + 3 a^{2} b + 3 a b^{2} + b^{3} - a^{3} + 3 (a^{2} b) - 3 (a b^{2}) + b^{3})$

Step 1.4.2.6

Remove parentheses.

$4 (a^{3} + 3 a^{2} b + 3 a b^{2} + b^{3} - a^{3} + 3 a^{2} b - 3 a b^{2} + b^{3})$

Step 1.4.3

Combine the opposite terms in $a^{3} + 3 a^{2} b + 3 a b^{2} + b^{3} - a^{3} + 3 a^{2} b - 3 a b^{2} + b^{3}$ .

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Step 1.4.3.1

Subtract $a^{3}$ from $a^{3}$ .

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

Step 1.4.3.2

Add $3 a^{2} b + 3 a b^{2} + b^{3}$ and $0$ .

$4 (3 a^{2} b + 3 a b^{2} + b^{3} + 3 a^{2} b - 3 a b^{2} + b^{3})$

Step 1.4.3.3

Subtract $3 a b^{2}$ from $3 a b^{2}$ .

$4 (3 a^{2} b + b^{3} + 3 a^{2} b + 0 + b^{3})$

Step 1.4.3.4

Add $3 a^{2} b + b^{3} + 3 a^{2} b$ and $0$ .

$4 (3 a^{2} b + b^{3} + 3 a^{2} b + b^{3})$

Step 1.4.4

Add $3 a^{2} b$ and $3 a^{2} b$ .

$4 (b^{3} + 6 a^{2} b + b^{3})$

Step 1.4.5

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

$4 (2 b^{3} + 6 a^{2} b)$

Step 1.4.6

Apply the distributive property.

$4 (2 b^{3}) + 4 (6 a^{2} b)$

Step 1.4.7

Multiply $2$ by $4$ .

$8 b^{3} + 4 (6 a^{2} b)$

Step 1.4.8

Multiply $6$ by $4$ .

$f' (a) = 8 b^{3} + 24 a^{2} b$

Step 2

Find the second derivative.

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Step 2.1

Differentiate.

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Step 2.1.1

By the Sum Rule, the derivative of $8 b^{3} + 24 a^{2} b$ with respect to $a$ is $\frac{d}{d a} [8 b^{3}] + \frac{d}{d a} [24 a^{2} b]$ .

$\frac{d}{d a} [8 b^{3}] + \frac{d}{d a} [24 a^{2} b]$

Step 2.1.2

Since $8 b^{3}$ is constant with respect to $a$ , the derivative of $8 b^{3}$ with respect to $a$ is $0$ .

$0 + \frac{d}{d a} [24 a^{2} b]$

Step 2.2

Evaluate $\frac{d}{d a} [24 a^{2} b]$ .

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Step 2.2.1

Since $24 b$ is constant with respect to $a$ , the derivative of $24 a^{2} b$ with respect to $a$ is $24 b \frac{d}{d a} [a^{2}]$ .

$0 + 24 b \frac{d}{d a} [a^{2}]$

Step 2.2.2

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

$0 + 24 b (2 a)$

Step 2.2.3

Multiply $2$ by $24$ .

$0 + 48 b a$

Step 2.3

Simplify.

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Step 2.3.1

Add $0$ and $48 b a$ .

$48 b a$

Step 2.3.2

Reorder the factors of $48 b a$ .

$f'' (a) = 48 a b$

Step 3

Find the third derivative.

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Step 3.1

Since $48 b$ is constant with respect to $a$ , the derivative of $48 a b$ with respect to $a$ is $48 b \frac{d}{d a} [a]$ .

$48 b \frac{d}{d a} [a]$

Step 3.2

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

$48 b \cdot 1$

Step 3.3

Multiply $48$ by $1$ .

$f''' (a) = 48 b$

Step 4

Since $48 b$ is constant with respect to $a$ , the derivative of $48 b$ with respect to $a$ is $0$ .

$f^{4} (a) = 0$