Calculus Examples

$P (x) = (x - 2) (3 x - 1)$

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

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

Step 2

Differentiate.

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Step 2.1

By the Sum Rule, the derivative of $3 x - 1$ with respect to $x$ is $\frac{d}{d x} [3 x] + \frac{d}{d x} [- 1]$ .

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

Step 2.2

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

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

Step 2.3

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

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

Step 2.4

Multiply $3$ by $1$ .

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

Step 2.5

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

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

Step 2.6

Simplify the expression.

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Step 2.6.1

Add $3$ and $0$ .

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

Step 2.6.2

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

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

Step 2.7

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

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

Step 2.8

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

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

Step 2.9

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

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

Step 2.10

Simplify the expression.

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Step 2.10.1

Add $1$ and $0$ .

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

Step 2.10.2

Multiply $3 x - 1$ by $1$ .

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

Step 3

Simplify.

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Step 3.1

Apply the distributive property.

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

Step 3.2

Combine terms.

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Step 3.2.1

Multiply $3$ by $- 2$ .

$3 x - 6 + 3 x - 1$

Step 3.2.2

Add $3 x$ and $3 x$ .

$6 x - 6 - 1$

Step 3.2.3

Subtract $1$ from $- 6$ .

$6 x - 7$