Derivative of product and quotient 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 product of two functions g(x) and h(x) then
y = f(x) = g(x) h(x)
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) h(x + δx) - g(x)h(x + δx) + g(x)h(x + δx) - g(x)h(x)
f(x + δx) - f(x) = [g(x + δx) - g(x)]h(x + δx) + g(x)[h(x + δx) - h(x)]

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

dy/dx = lim [f(x + δx) - f(x)]/δx
       δx→0
= lim {[g(x + δx) - g(x)]/δx} h(x + δx) + g(x){[h(x + δx) - h(x)]/δx}
  δx→0
= lim {[g(x + δx) - g(x)]/δx} h(x + δx) + g(x)lim {[h(x + δx) - h(x)]/δx}
  δx→0                                        δx→0
= [dg/dx]h(x) + g(x)[dh/dx]

df/dx = = [dg/dx]h + g[dh/dx] 


If f(x) is quotient of two functions g(x) and h(x) then
y = f(x) = g(x)/h(x) and h(x)≠0
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)h(x) - h(x + δx)g(x)]/h(x + δx)h(x)
f(x + δx) - f(x) = [g(x + δx)h(x) - g(x)h(x) - h(x + δx)g(x) + g(x)h(x)]/h(x + δx)h(x)
f(x + δx) - f(x) = {[g(x + δx) - g(x)]h(x) - [h(x + δx) - h(x)]g(x)}/h(x + δx)h(x)

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

dy/dx = lim [f(x + δx) - f(x)]/δx
       δx→0
= lim {[g(x + δx) - g(x)]/δx}{h(x)/h(x + δx)h(x)} 
  δx→0
- {[h(x + δx) - h(x)]/δx}{g(x)/h(x + δx)h(x)}
= lim {[g(x + δx) - g(x)]/δx} {h(x)/h(x + δx)h(x)} 
  δx→0     
- lim {[h(x + δx) - h(x)]/δx}{g(x)/h(x + δx)h(x)}
  δx→0
= {[dg/dx]h(x) - g(x)[dh/dx]}/[h(x)]2

df/dx = = {[dg/dx]h - g[dh/dx]}/h2


df/dx = = [dg/dx]h + g[dh/dx] when f =gh
df/dx = = {[dg/dx]h - g[dh/dx]}/h2 when h≠0 and f = g/h



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