If f(x) = 2a|3x - 9| - ax, where a is some constant not equal to zero, find f′(3)?

Solution:

Given: Function f(x) = 2a|3x - 9| - ax and a is some constant not equal to zero

We know that f'(x) is the derivative of f(x)

 f'(x) = d/dx [2a × |3x - 9| - ax ]

We have the chain rule of derivative,

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

We know that d/dx of |x| = x / |x|

f'(x) = 2a × (3x - 9) / |3x - 9| × (3 - a)

f'(x) = 6a × (3x - 9) / |3x - 9| - a --- (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)

Here, the denominator is 0, which is called an invalid fraction.

Thus, the derivative of f(x) at x = 3 is not possible.

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If f(x) = 2a|3x - 9| - ax, where a is some constant not equal to zero, find f′(3)?

Summary:

If f(x) = 2a|3x - 9| - ax, where a is some constant not equal to zero, then f′(3) doesn’t exist.