Question

If f(x) = 3a|4x - 4| - ax, where a is some constant not equal to zero, find f '(1)

Answer 1

the slope of an absolute function at the point where it evaluates to zero is undefined.
Here, when x=1, so |4x-4|=|4-4|=|0|, the slope is undefined. This is where the vertex of the "V" on the graph of the absolute value function.

Answer 2

Differential does not exist.

Explanation:

given function,

f(x) = 3 a|4 x - 4| - ax

opening the function

f(x) = 3 a(4 x - 4) - ax and f(x) = - 3 a(4 x - 4) - ax

f'(x) = 12 a - a and f(x) = -12 a - a

f'(x) = 11 a and f(x) = -13 a

f'(1⁺) = 11 a and f(1⁻) = -13 a

hence,

• f'(1⁺) ≠ f(1⁻)
• so, Differential does not exist.