#### To determine

**To find:** The derivative of
f(x). That is,
f′(x) and compare their graphs.

#### Answer

The derivative of
f(x) is
−x2−1(x2−1)2.

#### Explanation

**Given:**

The function
f(x)=x(x2−1).

**Derivative rules:**

(1) Constant Multiple Rule:
ddx[c⋅f(x)]=c⋅ddxf(x)

(2) Power Rule:
ddx(xn)=nxn−1

(3) Quotient Rule: If
f1(t) and
f2(t) are both differentiable, then

ddt[f1(t)f2(t)]=f2(x)ddt[f1(t)]−f1(t)ddt[f2(t)][f2(t)]2

(4) Difference Rule:
ddx[f(x)−g(x)]=ddx(f(x))−ddx(g(x))

**Calculation:**

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

f′(x)=ddx(f(x)) =ddx(x(x2−1))

Apply the rule (3), (1) and (4),

f′(x)=(x2−1)ddx(x)−(x)ddx((x2−1))(x2−1)2=(x2−1)(1)−(x)ddx(x2)+0(x2−1)2=(x2−1)(1)−(x)2x(x2−1)2=(x2−1−2x2)(x2−1)2

Simplify further as,

f′(x)=(x2−1−2x2)(x2−1)2=−x2−1(x2−1)2

Therefore, the derivative of the function
f(x) is
−x2−1(x2−1)2.

To compare the graphs of the functions
f(x) and
f′(x).

The functions are
f(x)=x(x2−1)and f′(x)= −x2−1(x2−1)2.

**Graph:**

The graph of the functions
f(x) and
f′(x) is shown below in Figure 1.

**Observation**:

From Figure 1, it is noticed that the dotted lines represents the derivative and black line represents the function,

(i) The tangent of
f(x) is always a negative value.

(ii) The graph of
f′(x) tends to zero when the derivative graphs goes to the right or left and the function flattens as it goes to the left or right.

(iii) The graph of
f′(x) is completely below the *x* axis, which means that it has a negative *y* value.

Thus, it can be concluded that the derivative of the function
f(x)=x(x2−1) is reasonable.