Calculus Examples

$f (x) = (4 x + 7) (x^{3} - 2)$

Step 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 $f (x) = 4 x + 7$ and $g (x) = x^{3} - 2$ .

$(4 x + 7) \frac{d}{d x} [x^{3} - 2] + (x^{3} - 2) \frac{d}{d x} [4 x + 7]$

Step 2

Differentiate.

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Step 2.1

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

$(4 x + 7) (\frac{d}{d x} [x^{3}] + \frac{d}{d x} [- 2]) + (x^{3} - 2) \frac{d}{d x} [4 x + 7]$

Step 2.2

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

$(4 x + 7) (3 x^{2} + \frac{d}{d x} [- 2]) + (x^{3} - 2) \frac{d}{d x} [4 x + 7]$

Step 2.3

Since $- 2$ is constant with respect to $x$ , the derivative of $- 2$ with respect to $x$ is $0$ .

$(4 x + 7) (3 x^{2} + 0) + (x^{3} - 2) \frac{d}{d x} [4 x + 7]$

Step 2.4

Add $3 x^{2}$ and $0$ .

$(4 x + 7) (3 x^{2}) + (x^{3} - 2) \frac{d}{d x} [4 x + 7]$

Step 2.5

By the Sum Rule, the derivative of $4 x + 7$ with respect to $x$ is $\frac{d}{d x} [4 x] + \frac{d}{d x} [7]$ .

$(4 x + 7) (3 x^{2}) + (x^{3} - 2) (\frac{d}{d x} [4 x] + \frac{d}{d x} [7])$

Step 3

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

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Step 3.1

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

$(4 x + 7) (3 x^{2}) + (x^{3} - 2) (4 \frac{d}{d x} [x] + \frac{d}{d x} [7])$

Step 3.2

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

$(4 x + 7) (3 x^{2}) + (x^{3} - 2) (4 \cdot 1 + \frac{d}{d x} [7])$

Step 3.3

Multiply $4$ by $1$ .

$(4 x + 7) (3 x^{2}) + (x^{3} - 2) (4 + \frac{d}{d x} [7])$

Step 4

Differentiate using the Constant Rule.

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Step 4.1

Since $7$ is constant with respect to $x$ , the derivative of $7$ with respect to $x$ is $0$ .

$(4 x + 7) (3 x^{2}) + (x^{3} - 2) (4 + 0)$

Step 4.2

Add $4$ and $0$ .

$(4 x + 7) (3 x^{2}) + (x^{3} - 2) \cdot 4$

Step 5

Simplify.

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Step 5.1

Apply the distributive property.

$4 x (3 x^{2}) + 7 (3 x^{2}) + (x^{3} - 2) \cdot 4$

Step 5.2

Apply the distributive property.

$4 x (3 x^{2}) + 7 (3 x^{2}) + x^{3} \cdot 4 - 2 \cdot 4$

Step 5.3

Combine terms.

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Step 5.3.1

Multiply $3$ by $4$ .

$12 x \cdot x^{2} + 7 (3 x^{2}) + x^{3} \cdot 4 - 2 \cdot 4$

Step 5.3.2

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

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Step 5.3.2.1

Move $x^{2}$ .

$12 (x^{2} x) + 7 (3 x^{2}) + x^{3} \cdot 4 - 2 \cdot 4$

Step 5.3.2.2

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

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Step 5.3.2.2.1

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

$12 (x^{2} x^{1}) + 7 (3 x^{2}) + x^{3} \cdot 4 - 2 \cdot 4$

Step 5.3.2.2.2

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

$12 x^{2 + 1} + 7 (3 x^{2}) + x^{3} \cdot 4 - 2 \cdot 4$

Step 5.3.2.3

Add $2$ and $1$ .

$12 x^{3} + 7 (3 x^{2}) + x^{3} \cdot 4 - 2 \cdot 4$

Step 5.3.3

Multiply $3$ by $7$ .

$12 x^{3} + 21 x^{2} + x^{3} \cdot 4 - 2 \cdot 4$

Step 5.3.4

Move $4$ to the left of $x^{3}$ .

$12 x^{3} + 21 x^{2} + 4 \cdot x^{3} - 2 \cdot 4$

Step 5.3.5

Multiply $- 2$ by $4$ .

$12 x^{3} + 21 x^{2} + 4 x^{3} - 8$

Step 5.3.6

Add $12 x^{3}$ and $4 x^{3}$ .

$16 x^{3} + 21 x^{2} - 8$