3.7: The Chain Rule

I am not a fan of the chain rule. It's super confusing when there's a lot going on in the function. The double chain rule is even worse..

Problem:

1. Find the derivative of: (x^3 + sec(ln5x) + sin(4^x) + e^x + 9x) / (cos(4x* 4th root of x))

First you use the quotient rule: (cos(4x*4th root of x)) * (x^3 + sec(ln5x) + sin(4x) + e^x + 9x)' - (x^3 + sec(ln5x) + sin(4^x) + e^x +9x) * (cos 4x*4th root of x)' ALL divided by (cos(4x*4th root of x))^2

Using a whole combination of rules, including the chain rule, the derivative of this function is:

cos(4x*4th root of x) * (3x^2) + sec(ln5x)*tan(ln5x)*(1/5x)*(5) + cos(4x)(ln4*4^x)+(e^x)+(9) - (x^3) + sec(ln5x)) + sin(4^x) + (e^x) +(9x) (-sin(4x*4th root of x)* (4)*((1/4)x^(-3/4)) ALL divided by (cos(4x*4th root of x))^2

I hope you can follow this, it looks crazy..