In calculus, the constant factor rule in differentiation allows one to take constants outside a derivative and concentrate on differentiating the function of x itself. This is a part of the linearity of differentiation.
Consider a differentiable function
where k is a constant.
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, 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.