#### To determine

The proof of the Quotient Rule.

#### Answer

The differentiation of the function is
A′(v)=23v−73(8v4+v2+2).

#### Explanation

**Given:**

The assumptions made are that
F′(x) exists, where
F=fg.

The product rule if applied to the function
f=Fg.

**Derivative rule:**

**(**1) Product Rule: If
f1(v) and
f2(v) are both differentiable, then

ddv[f1(v)f2(v)]=f1(v)ddv[f2(v)]+f2(v)ddv[f1(v)].

(2) Constant multiple rule:
ddx(cf)=cddx(f)

(3) Power rule:
ddx(xn)=nxn−1

(4) Sum rule:
ddx(f+g)=ddx(f)+ddx(g)

(5) Difference rule:
ddx(f−g)=ddx(f)−ddx(g)

**Calculation:**

Use product rule and differentiate
f=Fg as follows.

ddx(f)=Fg′+F′g

Subtract both sides by the function
Fg′ as,

f′−Fg′=Fg′+F′g−Fg′=F′g

Divide *g* throughout the equation as,

f′−Fg′g=F′ggf′−Fg′g=F′

Substitute
F=fg in the above equation as follows.

F′=f′−(fg)g′g=f′g−fg′gg=f′g−fg′g2

Hence, the quotient rule is
F′=f′g−fg′g2.