Calculus Examples

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

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

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

Step 2

Differentiate.

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Step 2.1

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

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

Step 2.2

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

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

Step 2.3

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

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

Step 2.4

Simplify the expression.

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Step 2.4.1

Add $1$ and $0$ .

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

Step 2.4.2

Multiply $x - 1$ by $1$ .

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

Step 3

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

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

Step 4

Differentiate.

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Step 4.1

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

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

Step 4.2

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

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

Step 4.3

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

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

Step 4.4

Simplify the expression.

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Step 4.4.1

Add $1$ and $0$ .

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

Step 4.4.2

Multiply $x - 1$ by $1$ .

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

Step 4.5

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

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

Step 4.6

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

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

Step 4.7

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

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

Step 4.8

Simplify by adding terms.

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Step 4.8.1

Add $1$ and $0$ .

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

Step 4.8.2

Multiply $x - 2$ by $1$ .

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

Step 4.8.3

Add $x$ and $x$ .

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

Step 4.8.4

Subtract $2$ from $- 1$ .

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

Step 5

Simplify.

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Step 5.1