dx(f(x)g(x)=g(x) dx(f(x))+f(x) dx(g(x))

Question

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Answer 1

This is the product rule. If u=f(x) and v=g(x),

d(uv)/dx=udv/dx+vdu/dx=f(x)d(g(x))/dx+g(x)d(f(x))/dx.

PROOF

Let Δx be a small change in x producing small changes Δu and Δv in u and v.

u=f(x), u+Δu=f(x+Δx), (f(x+Δx)-f(x))/Δx=Δu/Δx is the slope caused by the changes in u and x.

similarly (g(x+Δx)-g(x))/Δx=Δv/Δx is the slope caused by the changes in v and x.

Now consider the product uv and the change in the product when x changes.

(u+Δu)(v+Δv)=uv+uΔv+vΔu+ΔuΔv. When Δu and Δv are small, ΔuΔv becomes negligible.

(u+Δu)(v+Δv)-uv=uΔv+vΔu is the change in the product uv when u+Δu=f(x+Δx) and v+Δv=g(x+Δx).

[(u+Δu)(v+Δv)-uv]/Δx=(uΔv+vΔu)/Δx=u(Δv/Δx)+v(Δu/Δx).

As Δx→0, Δv/Δx→dv/dx, Δu/Δx→du/dx. The differential is the limit as Δx approaches zero.

Hence d(uv)/dx=d(f(x)g(x))=udv/dx+vdu/dx=f(x)d(g(x))/dx+g(x)d(f(x))/dx.