Derivative of sum and difference of two functions.

We know the formula for finding the derivative
dy/dx = lim [f(x + δx) - f(x)]/δx
       δx→0
If f(x) is a sum of two functions g(x) and h(x)
i.e. f(x) = g(x) + h(x) then
f(x + δx) = g(x + δx) + h(x + δx)

Difference
f(x + δx) - f(x) = g(x + δx) + h(x + δx) - (g(x) + h(x))
f(x + δx) - f(x) = g(x + δx) - g(x) + h(x + δx) - h(x)

Ratio
[f(x + δx) - f(x)]/δx = [g(x + δx) - g(x) + h(x + δx) - h(x)]/δx
[f(x + δx) - f(x)]/δx = [g(x + δx) - g(x)]/δx + [h(x + δx) - h(x)]/δx

Limit as δx→0

lim [f(x + δx) - f(x)]/δx
δx→0
= lim [g(x + δx) - g(x)]/δx + lim[h(x + δx) - h(x)]/δx
  δx→0                        δx→0

As limit of sum of two functions is equal to the sum of the limits of two functions.
df/dx = dg/dx + dh/dx

If f(x) is a difference of two functions g(x) and h(x) i.e.
f(x) = g(x) - h(x) then
df/dx = dg/dx - dh/dx 

df/dx = dg/dx + dh/dx if f(x) = g(x) + h(x)
and
df/dx = dg/dx - dh/dx if f(x) = g(x) - h(x)



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