Expert: Paul Klarreich - 2/14/2008

Question
This has to do with the product rule, but asks to differentiate and then write the derivative in factored form.  I'm having a difficult time using the product rule when there are exponents outside of paranthesis also.

f(x)= (x^2+3)^3(6-x)^5

Answer
Questioner:   Christine
Category:  Calculus
Private:  No
 
Subject:  Calculus
Question:  This has to do with the product rule, but asks to differentiate and then write the derivative in factored form.  I'm having a difficult time using the product rule when there are exponents outside of paranthesis also.

f(x)= (x^2+3)^3(6-x)^5
....................................
Hi again, Christine,

If the expression is a product, then it matches the PRODUCT RULE:

D(uv) = u Dv + v Du.

So identify the pieces (the factors), work on them one at a time, then put it all together.

f(x) = (x^2+3)^3(6-x)^5 = uv, where:

u = (x^2+3)^3  and

v = (6-x)^5

Now differentiate each one independently.  [don't worry about the overall example while doing this:

u = (x^2+3)^3, needs the G.P.R. [remember the last answer?]

Du = 3(x^2 + 3)^2 (2x), the 2x being  D(x^2 + 3)

= 6x(x^2 + 3)^2
.........
v = (6-x)^5 needs th G.P.R.

Dv = 5(6 - x)^4 (-1), the -1 being, oh, you know this already.
= - 5(6 - x)^4

Now put it all back together.

D(uv) = u Dv + v Du.

write the pattern:

D(uv) = ()() + ()()

Fill in:

D(uv) = ((x^2+3)^3)(- 5(6 - x)^4) + ((6-x)^5)(6x(x^2 + 3)^2)

Take out as many common factors as you can:  (x^2 + 3)^2  and (6 - x)^4  and then see what is left:

D(uv) = (x^2 + 3)^2(6 - x)^4[ -(x^2 + 3) + 6x(x - 5) ]

Now remove () inside the [] and simplify.