The derivative of the given function is f'(x) = k f(x) where
.
<h3>What is the derivative of a function?</h3>
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

<h3>Calculation:</h3>
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) = 
= ![\lim_{h \to 0} \frac{f(x)[f(h) - 1]}{h}](https://tex.z-dn.net/?f=%5Clim_%7Bh%20%5Cto%200%7D%20%5Cfrac%7Bf%28x%29%5Bf%28h%29%20-%201%5D%7D%7Bh%7D)
From property (ii),
= k;
f'(x) = ![\lim_{h \to 0} \frac{f(x)[f(h) - 1]}{h}](https://tex.z-dn.net/?f=%5Clim_%7Bh%20%5Cto%200%7D%20%5Cfrac%7Bf%28x%29%5Bf%28h%29%20-%201%5D%7D%7Bh%7D)
= f(x). 
= f(x). k
= kf(x)
Therefore, f'(x) = k f(x); where f'(x) exists for all real numbers of x.
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