Question

Which of the following statements are true or false?Please give reasons for your answers. (i)The derivative of quotient of two functions is equal to the quotient of derivatives of two functions.

Answer 1

False

Explanation:

This statement is false.

We will start with just an example. Let
`f(x) = 2x`, `g(x) = x^2`, and `h(x) = g(x) / f(x) = 1/2 x`

This is asking if the following is true:
`h'(x) = (g'(x))/(f'(x))`

We can easily check this:
`h'(x) = 1/2`
`g'(x) = 2x`
`f'(x) = 2`
so
`(g'(x))/(f'(x)) = (2x)/2 = x ne 1/2`

We also know this from the existence of the quotient rule:
`d/dx (f(x)/g(x)) = (g(x)f'(x) - f(x) g'(x))/(g(x))^2`

which is obviously not `(g'(x)) / (f'(x))`.