Solving derivative- possible without quotient rule?

1. Oct 28, 2013

Persimmon

1. The problem statement, all variables and given/known data
find the following derivative
d/dx [g(x + 1)(√(2+ (x + 8)^(1/3))/(cos(tan(sin(tan(sin x))))]
at x = 0

2. Relevant equations

3. The attempt at a solution

I split the big long derivative into 3 functions:
a(x) = g(x + 1)
b(x) = √(2+ (x + 8)^(1/3))
c(x) = cos(tan(sin(tan(sin x))))

and got : (a'x*bx + b'x*ax)(cx)-c'x*px*qx)/cx^2
subbing in x = 0 that simplified into 2*g'(1)
but it was mentioned that it's possible to solve this without the quotient rule. I can't figure out how you would solve this derivative if you didn't use the quotient rule.

2. Oct 28, 2013

Staff: Mentor

Instead of dividing by cos(...), you can multiply by sec(...). That way you're set up to use the product rule.

BTW, your notation is not very helpful.
As written this looks like a' * x * b * x etc. To be clearer, write a'(x) * b(x) etc.

3. Oct 28, 2013

Persimmon

Sorry for the unclear notation. I was trying to make it clearer so there wouldn't be endless brackets but I guess it's even more confusing that way. Thank you for your help!