### 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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