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.