Question

If y = f(x)g(x), then dy/dx = f‘(x)g‘(x). If it is true, explain your answer. If false, provide a counterexample. True or False?

Answer 1

False

Explanation:

Take `f(x)=x^2+1,g(x)=x`
then

`f(x)g(x)=x^3+x`
then
`f'(x)*g'(x)=2x`
But

`(f(x)g(x))'=3x^2+1`
It must be
`(f(x)g(x))'=f'(x)*g(x)+f(x)*g'(x)`

Answer 2

The statement is false .

The product rule provides the correct formulation:

`y = f(x)g(x) => y = f(x)g'(x) + f'(x)g(x)`

Explanation:

We can readily disprove the given statement:

Consider:

`f(x)=x` and `g(x)=x`

Then differentiating wrt `x` we have:

`f'(x)=1` and `g'(x)=1 => f'(x)g'(x)=1`

And `y=f(x)g(x) =x^2 => dy/dx = 2x != f'(x)g'(x)`

And so By counterexample, the statement is false .

In fact the product rule provides the correct formulation:

`y = f(x)g(x) => y = f(x)g'(x) + f'(x)g(x)`