Calculus Examples

$f (x) = (2 x + 1) (3 x - 2)$

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 $f (x) = 2 x + 1$ and $g (x) = 3 x - 2$ .

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

Step 2

Differentiate.

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Step 2.1

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

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

Step 2.2

Since $3$ is constant with respect to $x$ , the derivative of $3 x$ with respect to $x$ is $3 \frac{d}{d x} [x]$ .

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

Step 2.3

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

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

Step 2.4

Multiply $3$ by $1$ .

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

Step 2.5

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

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

Step 2.6

Simplify the expression.

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Step 2.6.1

Add $3$ and $0$ .

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

Step 2.6.2

Move $3$ to the left of $2 x + 1$ .

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

Step 2.7

By the Sum Rule, the derivative of $2 x + 1$ with respect to $x$ is $\frac{d}{d x} [2 x] + \frac{d}{d x} [1]$ .

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

Step 2.8

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

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

Step 2.9

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

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

Step 2.10

Multiply $2$ by $1$ .

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

Step 2.11

Since $1$ is constant with respect to $x$ , the derivative of $1$ with respect to $x$ is $0$ .

$3 (2 x + 1) + (3 x - 2) (2 + 0)$

Step 2.12

Simplify the expression.

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Step 2.12.1

Add $2$ and $0$ .

$3 (2 x + 1) + (3 x - 2) \cdot 2$

Step 2.12.2

Move $2$ to the left of $3 x - 2$ .

$3 (2 x + 1) + 2 \cdot (3 x - 2)$

Step 3

Simplify.

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Step 3.1

Apply the distributive property.

$3 (2 x) + 3 \cdot 1 + 2 (3 x - 2)$

Step 3.2

Apply the distributive property.

$3 (2 x) + 3 \cdot 1 + 2 (3 x) + 2 \cdot - 2$

Step 3.3

Combine terms.

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Step 3.3.1

Multiply $2$ by $3$ .

$6 x + 3 \cdot 1 + 2 (3 x) + 2 \cdot - 2$

Step 3.3.2

Multiply $3$ by $1$ .

$6 x + 3 + 2 (3 x) + 2 \cdot - 2$

Step 3.3.3

Multiply $3$ by $2$ .

$6 x + 3 + 6 x + 2 \cdot - 2$

Step 3.3.4

Multiply $2$ by $- 2$ .

$6 x + 3 + 6 x - 4$

Step 3.3.5

Add $6 x$ and $6 x$ .

$12 x + 3 - 4$

Step 3.3.6

Subtract $4$ from $3$ .

$12 x - 1$