Product and Quotient Rule - Lecture

Greeting Students;

This is your instructor Mishal. Today we will explain another topic of our financial class. This topic is “Products and Quotient Rule”

Product and Quotient Rule

• ·      PRODUCT RULE

At times, we may encounter a situation where the product of the derivatives of two functions is not the same as when the functions are differentiated after being multiplied.

For example, ( 2x² ) ( 6x³ )

U(x) = 2x² 

V(x) = 6x³

Differentiating the above functions separately then multiplying:

( 4x )( 18x² )

 = 72 x³

Multiplying before differentiating:

= 18 x⁵

Differentiating with respect to x yields:

= 90 x⁴

Thus u‘ (x) v’(x) ≠  ( uv) ‘ x

This expression proves the fact that the derivative of a product is not a product of its derivatives.

This is where the product rule comes in handy. It states that:

If two functions u(x) and v(x) are differentiable and the product is differentiable then:

(uv) ‘ = u’v+ uv’

  

Example: 

Differentiate , ( 2x² ) ( 6x³ )

Solution

U(x) = 2x² 

V(x) = 6x³

(uv) ‘ = u’v  + uv’= 4x ( 6x³ ) + 18x² ( 2x² )

= 24x⁴ + 36x⁴

= 60x⁴

• ·        QUOTIENT RULE

It utilizes nearly the same principles, only that it deals with quotients

It states that:

If the two functions  f(x) and g(x) are differentiable (i.e. the derivative exist) then the quotient is differentiable then:

Example: 

3x2 / 2x

U= 3x2

V=2x