#### To determine

**To find:** The derivative of the function in two different ways.

#### Answer

The derivative of the function f(x) is f′(x)=−8x3+6x2−2x+1.

#### Explanation

**Given:**

The function f(x)=(1+2x2)(x−x2).

**Derivative rule:**

(1) Product Rule: ddx[f1(x)f2(x)]=f1(x)ddx[f2(x)]+f2(x)ddx[f1(x)]

(2) Power rule: ddx(xn)=nxn−1

(3) Sum rule: ddx(f+g)=ddx(f)+ddx(g)

(4) Difference rule: ddx(f−g)=ddx(f)−ddx(g)

(5) Constant multiplicative rule: ddx(c⋅f)=c⋅ddx(f)

**Calculation:**

**Method 1:**

Obtain the derivative of f(x) by using the product rule.

The derivative of the function f(x) is f′(x), which is obtained as follows,

f′(x)=ddx(f(x))=ddx((1+2x2)(x−x2))

Substitute 1+2x2 for f1(x) and x−x2 for f2(x) in the product rule (1),

f′(x)=(1+2x2)ddx(x−x2)+(x−x2)ddx(1+2x2)

Apply the derivative rule (3), (4) and (5),

f′(x)=(1+2x2)[ddx(x)−ddx(x2)]+(x−x2)[ddx(1)+ddx(2x2)]=(1+2x2)[ddx(x)−ddx(x2)]+(x−x2)[ddx(1)+2ddx(x2)]

Apply the power rule (2) and simplify the terms,

f′(x)=(1+2x2)(1−2x)+(x−x2)(4x)=1−2x+2x2−4x3+4x2−4x3=−8x3+6x2−2x+1

Therefore, the derivative of the function f(x) is f′(x)=−8x3+6x2−2x+1.

**Method 2:**

Obtain the derivative of f(x) by performing the multiplication first.

The given function can be expressed as follows,

f(x)=(1+2x2)(x−x2)=x−x2+2x3−2x4=−2x4+2x3−x2+x

The derivative of the function f(x)=−2x4+2x3−x2+x is f′(x), which is obtained as follows,

f′(x)=ddx(f(x))=ddx(−2x4+2x3−x2+x)

Apply the derivative rule (3), (4) and (5),

f′(x)=ddx(−2x4)+ddx(2x3)−ddx(x2)+ddx(x)=−2ddx(x4)+2ddx(x3)−ddx(x2)+ddx(x)

Apply the power rule (2) and simplify the terms,

f′(x)=−2(4x4−1)+2(3x3−1)−(2x2−1)+(1x1−1)=−8x3+6x2−2x+1

Therefore, the derivative of the function f(x) is f′(x)=−8x3+6x2−2x+1.

Hence, it can be concluded that the derivative of the function by both methods **1** and **2** are same.