Derivatives are used to find the rate of changes of a quantity with respect to the other quantity. By using the application of derivatives we can find the approximate change in one quantity with respect to the change in the other quantity. Assume we have a function y = f(x), which is defined in the interval [a, a+h], then the average rate of change in the function in the given interval is
(f(a + h)-f(a))/h
Now using the definition of derivative, we can write
f'(a)=limh→0f(a+h)−f(a)hf′(a)=limh→0f(a+h)−f(a)h
which is also the instantaneous rate of change of the function f(x) at a.
Now, for a very small value of h, we can write
f'(a) ≈ (f(a+h) − f(a))/h
or
f(a+h) ≈ f(a) + f'(a)h
This means, if we want to find the small change in a function, we just have to find the derivative of the function at the given point, and using the given equation we can calculate the change. Hence the derivative gives the instantaneous rate of change of a function within the given limits and can be used to find the estimated change in the function f(x) for the small change in the other variable(x).