Derivative of a Sum One of our examples of a general derivative formula was: (u + v)� (x) = u� (x) + v � (x). (Remember that by (u + v)(x) we mean u(x) + v(x).) In other words, the derivative of the sum of two functions is just the sum of their derivatives. We’ll now prove that this is true for any pair of functions u and v, provided that those functions have derivatives. Since we don’t know in advance what functions u and v are, we can’t use any speciﬁc information about the functions or the slopes of their graphs; all we have to work with is the formal deﬁnition of the derivative. When we apply the deﬁnition of the derivative to the function (u + v)(x) we get: (u + v)(x + Δx) − (u + v)(x) (u + v)� (x) = lim Δx→0 Δx Since (u + v)(x) is just u(x) + v(x), u(x + Δx) + v(x + Δx) − u(x) − v(x) . Δx (u + v)� (x) = lim Δx→0 Combining like terms, we see that: (u + v)� (x) = lim u(x + Δx) − u(x) + v(x + Δx) − v(x) Δx � u(x + Δx) − u(x) v(x + Δx) − v(x) + Δx Δx Δx→0 or: � (u + v) (x) = lim Δx→0 � . Because u and v are diﬀerentiable (and therefore continuous), the limit of the sum is the sum of the limits. Therefore: u(x + Δx) − u(x) v(x + Δx) − v(x) + lim . Δx→0 Δx→0 Δx Δx (u + v)� (x) = lim The two limits above match the deﬁnition of the derivatives of u and v, so we’ve shown that (u + v)� (x) = u� (x) + v � (x).