#### To determine

**To find:** The first and second derivative of the function f(x)=3x2+2x+1.

#### Answer

The first and second derivatives of the function are, f′(x)=6x+2 and f″(x)=6.

#### Explanation

**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 derivative of the function f(x).

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

f′(x)=limh→0f(x+h)−f(x)h=limh→03(x+h)2+2(x+h)+1−{3x2+2x+1}h=limh→0(3x2+3h2+6xh)+(2x+2h)+1−{3x2+2x+1}h=limh→03x2+3h2+6xh+2x+2h+1−3x2−2x−1h

Simplify the numerator and to obtain the value of f′(x),

f′(x)=limh→03h2+6xh+2hh=limh→0h(3h+6x+2)h

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

f′(x)=limh→0(3h+6x+2)=3(0)+6x+2=6x+2

Thus, the first derivative of the function is, f′(x)=6x+2.

Obtain the second derivative of the function.

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

f′′(x) =(f′(x))′=limh→0f′(x+h)−f′(x)h=limh→06(x+h)+2−{6x+2}h=limh→06x+6h+2−6x−2h

Simplify the numerator and to obtain the derivatives f″(x),

f″(x) =limh→06hh

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

f″(x)=limh→0(6)=6

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

**To check:** The derivatives f(x), f′(x),.and f″(x) are reasonable by comparing the graphs of f(x), f′(x). and f″(x).

The derivative f(x), f′(x). and f″(x) are reasonable.

**Graph:**

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 straight line and the graph of f″(x) is a horizontal line.

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

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

The f′(x) represents a linear function and f″(x) represents a constant function.

The straight line has same slope at all points. That is, f″(x) is constant value.

From the Figure 1, the estimated slope of the function f′(x) is 6 which is same as the derivative of f′(x). That is, f″(x)=6.

Therefore, the function f″(x) is reasonable.