Question

if f(x) is increasing , is f'(x) also increasing?

Answer 1

no.
if f(x) increasing, f'(x) > 0

Answer 2

could be increasing or decreasing or neither

Answer 3

Let's run some tests.

\[f(x)=x^2,\]\[f'(x)=2x.\]Both functions are increasing.

\[f(x)=\sqrt{x},\]\[f'(x)=\frac{1}{2\sqrt{x}}.\]One function is increasing whilst the other is decreasing.

\[f(x)=2x,\]\[f'(x)=2.\]One function is increasing whilst the other remains constant.

Yup.

Answer 4

This is actually pretty clear if you think MVT, the answer is YES.