compute (x-1)(x+2)(x-3) then differentiate the result

⚠ (x-1)(x+2)(x-3) is read as (x-1)*(x+2)*(x-3): use (...)*(...) for multiplication, use (...)^(...) for a power

Let's evaluate:

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

ⓘ Multiply each term of the sum by the other factor, using the distributive law:

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

x \cdot x

+ x \cdot 2

- 1 \cdot x

- 1 \cdot 2

(x \cdot x

+ x \cdot 2

- 1 \cdot x

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

ⓘ Multiplying powers with the same base adds the exponents:

x \cdot x

x^{2}

■Multiplication by 1 preserves the original value:

1 \cdot x

x

■Multiplications are done before additions and subtractions:

1 \cdot 2

2

(x^{2}

+ x \cdot 2

- x

- 2) \cdot (x - 3)

ⓘ Group like terms by adding their coefficients:

x \cdot 2 - x

x

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

ⓘ Multiply each term of the sum by the other factor, using the distributive law:

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

x^{2} x

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

+ x \cdot x

+ x \cdot (-3)

- 2 x

- 2 \cdot (-3)

x^{2} x

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

+ x \cdot x

+ x \cdot (-3)

- 2 x

- 2 \cdot (-3)

ⓘ To multiply powers with the same base, add their exponents:

x^{2} x

x^{3}

x \cdot x

x^{2}

■The minus sign is placed in front of the product:

x^{2} \cdot (-3)

- 3 x^{2}

■Write the minus sign in front of the product:

x \cdot (-3)

- 3 x

■Perform multiplications before additions and subtractions. Two minus signs cancel out, the product of two negative numbers is positive:

-2 \cdot (-3)

6

x^{3}

- 3 x^{2}

+ x^{2}

- 3 x

- 2 x

+ 6

ⓘ Group like terms by adding their coefficients:

- 3 x^{2} + x^{2}

-2 x^{2}

■Like terms are grouped by adding their coefficients:

- 3 x - 2 x

-5 x

▶Result:

x^{3}

- 2 x^{2}

- 5 x

+ 6

Let's perform the differentiation:

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

ⓘ Differentiate a sum by differentiating each term, then adding the derivatives:

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

\frac{d}{d x} (x^{3})

+ \frac{d}{d x} (- 2 x^{2})

+ \frac{d}{d x} (- 5 x)

+ 0

\frac{d}{d x} (x^{3})

+ \frac{d}{d x} (- 2 x^{2})

+ \frac{d}{d x} (- 5 x)

+ 0

ⓘ The derivative of

x^{n}

n x^{n - 1}

\frac{d}{d x} (x^{3})

3 x^{2}

\frac{d}{d x} (x^{2})

2 x

■A constant factor stays in front of the derivative: that of

k u

k \cdot u'

\frac{d}{d x} (- 2 x^{2})

-2 \cdot \frac{d}{d x} (x^{2})

\frac{d}{d x} (- 5 x)

-5

■Adding 0 changes nothing:

\frac{d}{d x} (- 5 x) + 0

\frac{d}{d x} (- 5 x)

3 x^{2} - 2 \cdot 2 x - 5

ⓘ Perform multiplications before additions and subtractions:

2 \cdot 2

4

▶Derivative:

3 x^{2} - 4 x - 5