derivative of x^2(3-x) — step-by-step solution

1) Let's differentiate:

\frac{d}{d x} (x^{2} \cdot (3 - x))

We differentiate

u1

x^{2}

and

v2

3 - x

using:

\frac{d}{d x} (u1 \cdot v2) = \frac{d}{d x} (u1) \cdot v2 + u1 \cdot \frac{d}{d x} (v2)

2) Let's perform the differentiation

u1

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

ⓘ The derivative of x^n is n*x^(n-1):

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

2 x

2 x

3) Let's differentiate

v2

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

ⓘ The derivative of a sum is the sum of the derivatives:

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

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

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

ⓘ Adding 0 changes nothing:

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

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

■The derivative of x is 1:

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

-1

-1

4) We evaluate the resulting derivative (

u1 \cdot v2

)':

2 x \cdot (3 - x) + x^{2} \cdot (-1)

ⓘ Multiplying by -1 gives the opposite:

x^{2} \cdot (-1)

- x^{2}

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

ⓘ Expand by multiplying each term inside the parentheses:

2 x \cdot (3 - x)

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

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

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

2 \cdot (- x)

-2 x

6 x - 2 \cdot x \cdot x - x^{2}

ⓘ Multiplying powers with the same base adds the exponents:

x \cdot x

x^{2}

6 x - 2 x^{2} - x^{2}

ⓘ Like terms are grouped by adding their coefficients:

- 2 x^{2} - x^{2}

-3 x^{2}

6 x - 3 x^{2}

5) Then you can continue the derivation:

ⅰ

\frac{d}{d x} (x^{2} \cdot (3 - x))

6 x - 3 x^{2}

6 x - 3 x^{2}

You can write the expression in factored form:

x \cdot (6 - 3 x)

Let's factor in

ℝ

(6 - 3 x)

We factor out

3

, the greatest common divisor of the coefficients:

6 - 3 x

3 \cdot (2 - x)

Factor found:

2 - x

▶Derivative:

3 x \cdot (2 - x)