If f(x) = 2a|3x – 9| – ax, where a is some constant not equal to zero, find f ′(3).
Algebra is the branch of mathematics, which deals with variables and expressions. It has an immense number of applications and are used to find values of various parameters and quantities.
Answer: For f(x) = 2a|3x – 9| – ax, f'(3) does not exists.
Let's understand how we arrived at the solution.
We have f(x) = 2a|3x – 9| – ax.
Now, f'(x) = d/dx [2a × |3x - 9| - ax ], where f'(x) is the derivative of f'(x).
Using the chain rule of derivative we get,
f ' (x) = d/dx [ 2a × |3x - 9| - ax ]
f ' (x) = d/dx [ 2a × |3x - 9| ] - d/dx [ ax ]
f ' (x) = 2a × |3x-9| / (3x-9) × d/dx [ 3x - 9] - a [ Since, d/dx of |x| = |x| / x]
f ' (x) = 2a × |3x-9| / (3x-9) × 3 - a
f ' (x) = 6a × |3x-9| / (3x-9) - a
Now, to find f'(3), we substitute x = 3 in the above equation.
⇒ f'(3) = 6a|3(3) - 9| / (3(3) - 9) - a = 6a|9 - 9| / (9 - 9) - a = 6a |0| / 0 - a
Now, we arrive at a situation where the denominator is 0, which is not possible.