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The derivative of the given function is f'(x) = k f(x) where .
Let f be a function defined on a neighborhood of a real number a. Then f is said to be differentiable or derivable at 'a' if exists finitely. The limit is called the derivative or differential coefficient of f at 'a'. It is denoted by f'(a).
If f is differentiable at 'a', then
The given properties are:
(i) f(x + y) = f(x)f(y) for all real numbers x and y.
(ii) = k; where k is a nonzero real number.
Then, the derivative of the function f(x) is,
f'(x) =
From property (i), f(x + h) = f(x)f(h)
On substituting,
f'(x) =
=
From property (ii), = k;
f'(x) =
= f(x).
= f(x). k
= kf(x)
Therefore, f'(x) = k f(x); where f'(x) exists for all real numbers of x.
Learn more about the derivative of a function here:
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