Constant factor rule in differentiation

In calculus, the constant factor rule in differentiation allows you to take constants outside a derivative and concentrate on differentiating the function of x itself.

Suppose you have a function

Use the formula for differentiation from first principles to obtain:

This is the statement of the constant factor rule in differentiation, in Lagrange's notation for differentiation.

In Leibniz's notation for differentiation, this reads

If we put k=-1 in the constant factor rule for differentiation, we have:

Comment on proof

Note that for this statement to be true, k must be a constant, or else the k can't be taken outside the limit in the line marked (*).

If k depends on x there is no reason to think k(x+h) = k(x). In that case the more complicated proof of the product rule applies.