Algebra Examples

$(3 - c^{2} d) (4 - 4 c^{2} d)$

Step 1

Differentiate using the Product Rule which states that $\frac{d}{d c} [f (c) g (c)]$ is $f (c) \frac{d}{d c} [g (c)] + g (c) \frac{d}{d c} [f (c)]$ where $f (c) = 3 - c^{2} d$ and $g (c) = 4 - 4 c^{2} d$ .

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

Step 2

Differentiate.

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Step 2.1

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

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

Step 2.2

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

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

Step 3

Evaluate $\frac{d}{d c} [- 4 c^{2} d]$ .

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Step 3.1

Since $- 4 d$ is constant with respect to $c$ , the derivative of $- 4 c^{2} d$ with respect to $c$ is $- 4 d \frac{d}{d c} [c^{2}]$ .

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

Step 3.2

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

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

Step 3.3

Multiply $2$ by $- 4$ .

$(3 - c^{2} d) (0 - 8 d c) + (4 - 4 c^{2} d) \frac{d}{d c} [3 - c^{2} d]$

Step 4

Simplify.

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Step 4.1

Subtract $8 d c$ from $0$ .

$(3 - c^{2} d) (- 8 d c) + (4 - 4 c^{2} d) \frac{d}{d c} [3 - c^{2} d]$

Step 4.2

Reorder the factors of $- 8 d c$ .

$(3 - c^{2} d) (- 8 c d) + (4 - 4 c^{2} d) \frac{d}{d c} [3 - c^{2} d]$

Step 5

Differentiate.

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Step 5.1

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

$(3 - c^{2} d) (- 8 c d) + (4 - 4 c^{2} d) (\frac{d}{d c} [3] + \frac{d}{d c} [- c^{2} d])$

Step 5.2

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

$(3 - c^{2} d) (- 8 c d) + (4 - 4 c^{2} d) (0 + \frac{d}{d c} [- c^{2} d])$

Step 6

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

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Step 6.1

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

$(3 - c^{2} d) (- 8 c d) + (4 - 4 c^{2} d) (0 - d \frac{d}{d c} [c^{2}])$

Step 6.2

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

$(3 - c^{2} d) (- 8 c d) + (4 - 4 c^{2} d) (0 - d (2 c))$

Step 6.3

Multiply $2$ by $- 1$ .

$(3 - c^{2} d) (- 8 c d) + (4 - 4 c^{2} d) (0 - 2 d c)$

Step 7

Simplify.

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Step 7.1

Subtract $2 d c$ from $0$ .

$(3 - c^{2} d) (- 8 c d) + (4 - 4 c^{2} d) (- 2 d c)$

Step 7.2

Reorder the factors of $- 2 d c$ .

$(3 - c^{2} d) (- 8 c d) + (4 - 4 c^{2} d) (- 2 c d)$

Step 8

Simplify.

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Step 8.1

Apply the distributive property.

$3 (- 8 c d) - c^{2} d (- 8 c d) + (4 - 4 c^{2} d) (- 2 c d)$

Step 8.2

Apply the distributive property.

$3 (- 8 c d) - c^{2} d (- 8 c d) + 4 (- 2 c d) - 4 c^{2} d (- 2 c d)$

Step 8.3

Combine terms.

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Step 8.3.1

Multiply $- 8$ by $3$ .

$- 24 (c d) - c^{2} d (- 8 c d) + 4 (- 2 c d) - 4 c^{2} d (- 2 c d)$

Step 8.3.2

Multiply $- 8$ by $- 1$ .

$- 24 c d + 8 c^{2} d (c d) + 4 (- 2 c d) - 4 c^{2} d (- 2 c d)$

Step 8.3.3

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

$- 24 c d + 8 (c^{1} c^{2}) d \cdot d + 4 (- 2 c d) - 4 c^{2} d (- 2 c d)$

Step 8.3.4

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

$- 24 c d + 8 c^{1 + 2} d \cdot d + 4 (- 2 c d) - 4 c^{2} d (- 2 c d)$

Step 8.3.5

Add $1$ and $2$ .

$- 24 c d + 8 c^{3} d \cdot d + 4 (- 2 c d) - 4 c^{2} d (- 2 c d)$

Step 8.3.6

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

$- 24 c d + 8 c^{3} (d^{1} d) + 4 (- 2 c d) - 4 c^{2} d (- 2 c d)$

Step 8.3.7

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

$- 24 c d + 8 c^{3} (d^{1} d^{1}) + 4 (- 2 c d) - 4 c^{2} d (- 2 c d)$

Step 8.3.8

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

$- 24 c d + 8 c^{3} d^{1 + 1} + 4 (- 2 c d) - 4 c^{2} d (- 2 c d)$

Step 8.3.9

Add $1$ and $1$ .

$- 24 c d + 8 c^{3} d^{2} + 4 (- 2 c d) - 4 c^{2} d (- 2 c d)$

Step 8.3.10

Multiply $- 2$ by $4$ .

$- 24 c d + 8 c^{3} d^{2} - 8 (c d) - 4 c^{2} d (- 2 c d)$

Step 8.3.11

Multiply $- 2$ by $- 4$ .

$- 24 c d + 8 c^{3} d^{2} - 8 c d + 8 c^{2} d (c d)$

Step 8.3.12

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

$- 24 c d + 8 c^{3} d^{2} - 8 c d + 8 (c^{1} c^{2}) d \cdot d$

Step 8.3.13

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

$- 24 c d + 8 c^{3} d^{2} - 8 c d + 8 c^{1 + 2} d \cdot d$

Step 8.3.14

Add $1$ and $2$ .

$- 24 c d + 8 c^{3} d^{2} - 8 c d + 8 c^{3} d \cdot d$

Step 8.3.15

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

$- 24 c d + 8 c^{3} d^{2} - 8 c d + 8 c^{3} (d^{1} d)$

Step 8.3.16

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

$- 24 c d + 8 c^{3} d^{2} - 8 c d + 8 c^{3} (d^{1} d^{1})$

Step 8.3.17

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

$- 24 c d + 8 c^{3} d^{2} - 8 c d + 8 c^{3} d^{1 + 1}$

Step 8.3.18

Add $1$ and $1$ .

$- 24 c d + 8 c^{3} d^{2} - 8 c d + 8 c^{3} d^{2}$

Step 8.3.19

Subtract $8 c d$ from $- 24 c d$ .

$- 32 c d + 8 c^{3} d^{2} + 8 c^{3} d^{2}$

Step 8.3.20

Add $8 c^{3} d^{2}$ and $8 c^{3} d^{2}$ .

$- 32 c d + 16 c^{3} d^{2}$