#### To determine

**To find:** The first and second derivative of the function f(x)=x3−3x and sketch its graphs.

#### Answer

The first and second derivatives are, f′(x)=3x2−3 and f″(x)=6x.

#### Explanation

**Given:**

The function is, f(x)=x3−3x.

**Formula used:**

The derivative of a function *f ,* denoted by f′(x), is

f′(x)=limh→0f(x+h)−f(x)h (1)

**Calculation:**

Obtain the first derivative of the function f(x).

Compute f′(x) by using the equation (1),

f′(x)=limh→0f(x+h)−f(x)h=limh→0(x+h)3−3(x+h)−{x3−3x}h=limh→0x3+h3+3x2h+3xh2−3x−3h−x3+3xh

Simplify the numerator and obtain the derivative of the function,

f′(x)=limh→0h3+3x2h+3xh2−3hh=limh→0h(h2+3x2+3xh−3)h

Since the limit *h* approaches zero but is not equal to zero, cancel the common term *h* from both the numerator and the denominator,

f′(x)=limh→0(h2+3x2+3xh−3)=02+3x2+3x(0)−3=3x2−3

Thus, the first derivative of the function is, f′(x)=3x2−3.

Obtain the second derivative of the function f(x).

Compute f″(x) by using the equation (1),

f″(x)=(f′(x))′=limh→0f′(x+h)−f′(x)h=limh→03(x+h)2−3−{3x2−3}h=limh→03x2+3h2+6xh−3−3x2+3h

Simplify the numerator,

f″(x)=limh→03h2+6xhh=limh→0h(3h+6x)h

Since the limit *h* approaches zero but is not equal to zero, cancel the common term *h* from both the numerator and the denominator,

f″(x)=limh→0(3h+6x)=3(0)+6x=6x

Thus, the second derivative of the function is, f″(x)=6x.

Therefore, the first and second derivatives of the functions are, f′(x)=3x2−3 and f″(x)=6x.

Use the online graphing calculator to draw the graph of the functions f(x),f′(x) and f″(x) as shown below in Figure 1.

From Figure 1, it is observed that the graph of f′(x) is a curve and the graph of f″(x) is a straight line.

Take several points on the domain and estimate the slope of the function f(x) which is equal to the value of f′(x) at that point.

For example, take the point 1 and the value of the slope of the tangent to the function f(x) at 1 is 0 (horizontal tangent) which is same as the value f′(1)=0.

Thus, the derivative f′(x) is reasonable.

Take several points on the domain and estimate the slope of tangent to the function f′(x) which is equal to the value of f′′(x) at that point.

For example take the point 2 and the value of the slope of the tangent to the function f′(x) at 2 is 12 which is same as the value f″(2)=12.

Thus, the derivative f″(x) is reasonable.