Expert: Paul Klarreich - 10/26/2005

Question
find the value of (f * g)' at the given value of x.

1) f(u)=u^5+1, u=g(x)=sqrt(x), x=1
2) f(u)=cot(pi(u)/10), u=g(x)=(1)/(1-x), x=-1

Answer
Hi, Amy,
Like other chain rule examples, you write that:

dy   dy du
-- = -- --
dx   du du

After writing the two derivatives, you multiply and then eliminate u by substitution.  In these cases, of course, you can do something else:

Obtain the value of u for a given x, and substitute that:

For your (1):
Now at x = 1, u = 1, and
1. dy = u^5 + 1  -->  dy/du = 5u^4 = 5
2. u = sqrt(x), ---> du/dx = 1/2sqrt(x) = 1/2

So dy/dx 5(1/2) = 5/2

For your (2),
At x = -1, u = 1/2

y = cot(pi u/10) -->  

dy/du = - (pi/10) csc^2(pi u/10) =
     = - (pi/10) csc^2(pi (1/2/10)=
     = - (pi/10) csc^2(pi/20)= etc.

u = 1/(1-x) = (1-x)^-1 -->

du/dx = (1-x)^-2 = 1/4

Now multiply.