To determine

To show: The function f(x)=|x−6| is not differentiable at x=6 and to find the formula of the derivative of f(x).

Explanation

Definition used:

An absolute function |x| is defined as follows.

|x|={−x if x<0x if x≥0

Proof:

Rewrite the function f(x) in terms of a piecewise function as follows.

Use the definition of the absolute value function to define the function |x−6|,

f(x)={−(x−6)if x−6<0x−6if x−6≥0={−x+6if x<6x−6if x≥6

The left hand derivative at x=6 is calculated as follows,

f′−(6)=limx→6−f(x)−f(6)x−6

Since x tends to 6−, that is  x<6.

f′−(6)=limx→6−−x+6−(−6+6)x−6=limx→6−−x+6x−6=limx→6−−(x−6)(x−6)=−1

Thus, f′−(6)=−1 (1)

The right hand derivative at x=6 is calculated as follows,

f′+(6)=limx→6+f(x)−f(6)x−6

Since x tends to 6+, that is x>6.

f′+(6)=limx→6+x−6−(6−6)x−6=limx→6+x−6x−6=1

Thus, f′+(6)=1 (2)

From equations (1) and (2), f′−(6)≠f′+(6).

This implies that the value of f′(6) does not exist.

Therefore, the function f(x)={−x+6if x<6x−6if x≥6 is not differentiable at x=6

Hence, the required proof is obtained.

Now find the formula of derivative of f(x).

From equation (1) and (2), f′−(6)=−1 and f′+(6)=1.

This implies that, the function f′(x)=−1 at x<6 and f′(x)=1 at x>6.

Thus, the required derivative function is, f′(x)={−1if x<61if x>6.

Use the above information to draw the graph of f′ as shown below in Figure 1.

From Figure 1, it is observed that the function f′ is a step function.