The Chain Rule

I.  Quiz

II.  Homework Questions

III.  The Chain Rule

Our goal is to differentiate functions such as

y = (3x + 1)¹⁰  

The chain rule states that if y is a function of  u and u is a function of x then

dy/dx = (dy/du)(du/dx)

In our example we have

y = u¹⁰

and u = 3x + 1 

so that

dy/dx = (dy/du)(du/dx) = (10u⁹)(3) = 30(3x+1)⁹  

Proof of the chain rule:

Recall an alternate definition of the derivative:

Exercises:  find f'(x) if

A.  f(x) = (x³ - x + 1)²⁰

B.  f(x) = (x⁴ -3x³ + x)⁵

C.  f(x) = (1 - x)⁹ (1-x²)⁴

D.  f(x) = (x³ + 4x - 3)⁷/(2x - 1)³

E.  f(x)  = [x²(5 - x³)⁴]/(3 - x)