# Olivia gifted Sara a necklace

Olivia Sanchez was a friend of Sara. They were together in the middle school but then Olivia moved and had to go to a different school. A couple of years ago, she had visited Sara and then their friendship rekindled. This time, Olivia came to Sara and told her that she had just come from a vacation in Mexico. She had visited relatives, seen Copper Canyon, many ancient buildings and a lot of beautiful beaches. Of course, she had also done some shopping, and had brought a gift for Sara. It was a long chain necklace with a cactus pendant held by the two ends of the chain. Sara accepted the gift graciously and thanked Olivia. They chatted for a while catching up with the gossip after which Olivia left. Sara then went over to her boyfriend Johnny’s house.

Johnny was excited about the discussion of derivatives with Sara yesterday. He had even gone over parts of his Calculus book. He found some problems in the book to be easy. Even then he found a few questions which challenged him.

Sara: Hi Johnny. You told me that you were going to go over your Calculus book. Did you?

# Chain rule for differentiation

Johnny: Yes, I did. Many of the exercise problems were straight forward. I had a problem with some of them.

Sara: I did all of them. Which one did you have a problem with?

Johnny; Find f ^{‘}(x) when f(x) = sin (ax^{2} + bx ).

Sara: Oh. That’s simple. You can write m = ax^{2} + bx.

Then f(x) = sin m or df(x)/dm = cos m.

Now because m = ax^{2} + bx, dm/dx = 2ax + b, and, therefore,

f ‘(x) = df(x)/dx = df(x)/dm x dm/dx = cos m x (2ax+b) = (2ax +b) cos (2ax^{2}+bx).

Johnny: I didn’t know you could do that.

Sara: Yes, this is called the chain rule.

Then Sara just paused as if amused.

Johnny: Now what?

Sara: Olivia just gifted me a chain necklace that she brought from Mexico. How did she know that you were going to ask me about the chain rule today?

Johnny: You used a chain of two steps. I wonder if you can you have longer chains and do the same thing like dy/dx = dy/da . da/db .db/dc .dc/dx.

Sara: That’s the beauty of it. See how many links in the chain necklace she gave me. Chains can be as long as you want – even longer than your bike rides.

Johnny: Thanks for your help but quit making fun of my bike rides, please.

*Challenge *

If y = √cos(5x+1) find dy/dx.

*Solution*

y = √cos(5x+1) can also be written as y = (cos(5x+1))^{1/2}

Write u instead of cos (5x+1), then y = u^{1/2}

dy/du = 1/2 u^{-1/2}

In u = cos (5x+1), we can write v = 5x+1, then u = cos v and du/dv = – sin v

Since v = 5x+1, dv/dx = 5. Therefore, using chain rule

dy/dx = dy/du x du/dv x dv/dx = 1/2 u^{-1/2 }x (-sin v) x 5

or dy/dx = 1/2 (cos (5x+1)^{ -1/2} x (-sin (5x+1)) x 5 = – (cos (5x+1)^{ -1/2} x (sin (5x+1)) x 5/2.