#### To determine

**To find:** The derivative of f(x)=x2−x2.

#### Answer

The derivative of f(x)=x2−x2 is f′(x)=2−2x22−x2_.

#### Explanation

**Given:**

The function is f(x)=x2−x2.

**Result used:**

**The Power Rule combined with the Chain Rule:**

If *n* is any real number and g(x) is differentiable function, then

ddx[g(x)]n=n[g(x)]n−1g′(x) (1)

**Derivative Rule: Product Rule:**

If f(x). and g(x) are both differentiable function, then

ddx[f(x)g(x)]=f(x)ddx[g(x)]+g(x)ddx[f(x)] (2)

**Calculation:**

Obtain the derivative of f(x).

f′(x)=ddx(f(x))=ddx(x2−x2)

Apply the product rule as shown in equation (2),

f′(x)=xddx[2−x2]+2−x2ddx[x]=xddx(2−x2)12+2−x2[1x1−1]=xddx(2−x2)12+2−x2[x0]=xddx(2−x2)12+2−x2

Apply the power rule combined with the chain rule as shown in equation (1),

f′(x)=x12(2−x2)12−1ddx(2−x2)+2−x2=x12(2−x2)1−22(ddx(2)−ddx(x2))+2−x2=x12(2−x2)−12(0−(2x2−1))+2−x2=x12(2−x2)12(−2x1)+2−x2

Simplify further and obtain the derivative,

f′(x)=−x22−x2+2−x2=−x2+2−x22−x22−x2=−x2+2−x22−x2=2−2x22−x2

Therefore, the derivative of f(x)=x2−x2 is f′(x)=2(1−x2)2−x2_.

#### To determine

**To sketch:** The graph of f(x) and f′(x).

#### Explanation

**Given:**

The function is f(x)=x2−x2.

The derivative of f(x) is 2−2x22−x2.

**Graph:**

Use the online graphing calculator and draw the graph of the functions as shown below in Figure 1.

From Figure 1 it is observe that,

If f′(x) is positive then f(x) is increasing function.

If f′(x) is negative then f(x) is decreasing function.

If f(x) is local extrema (that is local minimum or local maximum), then f′(x) is crosses the *x* axis (f′(x)=0).

Hence, the function f′(x) is reasonable.