Calculus Examples

$f (x) = (6 x + 5) (x^{3} - 2)$

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 $f (x) = 6 x + 5$ and $g (x) = x^{3} - 2$ .

$(6 x + 5) \frac{d}{d x} [x^{3} - 2] + (x^{3} - 2) \frac{d}{d x} [6 x + 5]$

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 $\frac{d}{d x} [x^{3}] + \frac{d}{d x} [- 2]$ .

$(6 x + 5) (\frac{d}{d x} [x^{3}] + \frac{d}{d x} [- 2]) + (x^{3} - 2) \frac{d}{d x} [6 x + 5]$

Step 2.2

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

$(6 x + 5) (3 x^{2} + \frac{d}{d x} [- 2]) + (x^{3} - 2) \frac{d}{d x} [6 x + 5]$

Step 2.3

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

$(6 x + 5) (3 x^{2} + 0) + (x^{3} - 2) \frac{d}{d x} [6 x + 5]$

Step 2.4

Simplify the expression.

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Step 2.4.1

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

$(6 x + 5) (3 x^{2}) + (x^{3} - 2) \frac{d}{d x} [6 x + 5]$

Step 2.4.2

Move $3$ to the left of $6 x + 5$ .

$3 \cdot (6 x + 5) x^{2} + (x^{3} - 2) \frac{d}{d x} [6 x + 5]$

Step 2.5

By the Sum Rule, the derivative of $6 x + 5$ with respect to $x$ is $\frac{d}{d x} [6 x] + \frac{d}{d x} [5]$ .

$3 (6 x + 5) x^{2} + (x^{3} - 2) (\frac{d}{d x} [6 x] + \frac{d}{d x} [5])$

Step 2.6

Since $6$ is constant with respect to $x$ , the derivative of $6 x$ with respect to $x$ is $6 \frac{d}{d x} [x]$ .

$3 (6 x + 5) x^{2} + (x^{3} - 2) (6 \frac{d}{d x} [x] + \frac{d}{d x} [5])$

Step 2.7

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

$3 (6 x + 5) x^{2} + (x^{3} - 2) (6 \cdot 1 + \frac{d}{d x} [5])$

Step 2.8

Multiply $6$ by $1$ .

$3 (6 x + 5) x^{2} + (x^{3} - 2) (6 + \frac{d}{d x} [5])$

Step 2.9

Since $5$ is constant with respect to $x$ , the derivative of $5$ with respect to $x$ is $0$ .

$3 (6 x + 5) x^{2} + (x^{3} - 2) (6 + 0)$

Step 2.10

Simplify the expression.

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Step 2.10.1

Add $6$ and $0$ .

$3 (6 x + 5) x^{2} + (x^{3} - 2) \cdot 6$

Step 2.10.2

Move $6$ to the left of $x^{3} - 2$ .

$3 (6 x + 5) x^{2} + 6 \cdot (x^{3} - 2)$

Step 3

Simplify.

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Step 3.1

Apply the distributive property.

$(3 (6 x) + 3 \cdot 5) x^{2} + 6 (x^{3} - 2)$

Step 3.2

Apply the distributive property.

$3 (6 x) x^{2} + 3 \cdot 5 x^{2} + 6 (x^{3} - 2)$

Step 3.3

Apply the distributive property.

$3 (6 x) x^{2} + 3 \cdot 5 x^{2} + 6 x^{3} + 6 \cdot - 2$

Step 3.4

Combine terms.

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Step 3.4.1

Multiply $6$ by $3$ .

$18 x \cdot x^{2} + 3 \cdot 5 x^{2} + 6 x^{3} + 6 \cdot - 2$

Step 3.4.2

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

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Step 3.4.2.1

Move $x^{2}$ .

$18 (x^{2} x) + 3 \cdot 5 x^{2} + 6 x^{3} + 6 \cdot - 2$

Step 3.4.2.2

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

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Step 3.4.2.2.1

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

$18 (x^{2} x^{1}) + 3 \cdot 5 x^{2} + 6 x^{3} + 6 \cdot - 2$

Step 3.4.2.2.2

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

$18 x^{2 + 1} + 3 \cdot 5 x^{2} + 6 x^{3} + 6 \cdot - 2$

Step 3.4.2.3

Add $2$ and $1$ .

$18 x^{3} + 3 \cdot 5 x^{2} + 6 x^{3} + 6 \cdot - 2$

Step 3.4.3

Multiply $3$ by $5$ .

$18 x^{3} + 15 x^{2} + 6 x^{3} + 6 \cdot - 2$

Step 3.4.4

Multiply $6$ by $- 2$ .

$18 x^{3} + 15 x^{2} + 6 x^{3} - 12$

Step 3.4.5

Add $18 x^{3}$ and $6 x^{3}$ .

$24 x^{3} + 15 x^{2} - 12$