Finding a counterexample to wrong derivatives

[megadeth95]

Hello Forum, I find this problem to be extremely difficult. Any help is appreacited!

Each of the following "derivatives" has been done incorrectly. In each case below, find a counterexample, that shows why the reasoning is incorrect. Then correct the mistakes by finding the actual derivatives of the left hand side.


For example: The statement "d/dx f(2) = f'(2x) is false because if f(x)=x^2, then we have d/dx f(2x) = d/dx [4x^2] = 8x, but on the other hand, f'(x) = 2x, so f'(2x) = 4x . In general, the correct derivative would be "d/dx f(2x) = f'(2x) * 2" by the chain rule.


(a) d/dx f(x+5) = f'(x)
(b) d/dt g'(t/3) = g'(t) / 3


Attempts


Correcting the mistakes is easy but finding a counterexample is not. For question (a), I reasoned that the counterexample would be the anti-derivative of x+5, which is (1/2)x^2 + 5x , just like in the example they use x^2 (which is the anti-derivative of 2x) From there, this is what I did:


if f(x) = (1/2)x^2 + 5x


then --> d/dx f(x+5) = d/dx [ (x^2 + 20x +75)/2 ] = (4x+40)/4


but on the other hand f'(x) = x +5


so f'(x+5) = x + 5


therefore x + 5 ≠ (is not equal to) (4x+40)/4


--------------------------------------------------------------------


Similarly, I applied the same idea for question (b)


if g(t) = (1/6)t^2


then --> d/dt g(t/3) = d/dt [ (t^2)/54 ] = (108t)/54^2


but on the other hand f'(x) = t/3


so f'(t/3) = (36t)/18^2


therefore (36t)/18^2 ≠ (is not equal to) (108t)/54^2




I think these are not the counterexamples they are asking for. How is the right way to find them? Should I use other equations or formulas?


Thank you in advance for any help you might be able to provide.

Sincerely

megadeth95

 

[HallsofIvy]

I don't know what you mean by saying "these are not the counterexamples they are asking for." A counter-example is a counterexample! They asked for a counter-example, not a specific one. As long as the example you give shows that the property is NOT true, it is a valid counter-example.

 

[megadeth95]

I don't know what you mean by saying "these are not the counterexamples they are askingfor." A counter-example is a counterexample! They asked for a counter-example, not a specific one. As long as the example you give shows that the property is NOT true, it is a valid counter-example.

Hello, what i meant was if the counterexamples shown in my attempts were correct. Are they? thanks