Trigo 2:Chain Rule

Trigo 2:

Chain Rule

by Thavarajah Selvarajah

​For the following, decide whether

chain rule ( PIT-STOP 2) is needed?

​When

chain rule ( PIT-STOP 2) is needed?

Multiple Choice

Given $$y=\sin\left(1-3x\right)$$  ,then   $$\frac{d}{dx}\left[\sin\left(1-3x\right)\right]$$  =

• Yes

• No

Multiple Choice

Given $$y=\sin\left(3x\right)$$  ,then   $$\frac{d}{dx}\left(\sin3x\right)$$  =

• Yes

• No

Multiple Choice

Given $$y=3\sin\left(x\right)$$  ,then   $$\frac{d}{dx}\left(3\sin x\right)$$  =

• Yes

• No

Multiple Choice

Given $$y=3\sin\frac{2}{3}x$$  ,then   $$\frac{d}{dx}\left(3\sin\frac{2}{3}x\right)$$  =

• Yes

• No

Multiple Choice

Given $$y=3\sin w$$  ,then   $$\frac{d}{dw}\left(3\sin w\right)$$  =

• Yes

• No

​WENT WELL?

COULD YOU DISTINGUISH WHICH ONES NEEDS PIT-STOP 2

( CHAIN RULE) ?

LEVEL 2 : THE PROCESS of Chain Rule :) ​

Multiple Choice

What is  $$p?$$  

• $$\sin\ \left(1-3x\right)$$  

• $$-3x$$  

• $$1-3x$$  

• $$3x$$  

Multiple Choice

What is  $$q?$$  

• $$\sin\ u$$  

• $$\cos\ u$$  

• $$\sin\ x$$  

• $$\cos\ x$$  

Multiple Choice

What is  $$r?$$  

• $$\cos\ \left(1-3x\right)$$  

• $$-3\cos\ u$$  

• $$-3\cos\ \left(1-3x\right)$$  

• $$\cos\ x$$  

Multiple Choice

Given is 1st step of Chain Rule. What is $$a?$$  

• $$u$$  

• $$\tan\ 2x$$  

• $$2x$$  

Multiple Choice

Given is continuation step of previous Chain Rule. What is $$b$$  and  $$c?$$  

• $$b=2,\ c=\tan\ u$$

• $$b=-2,\ c=\sec^2u$$

• $$b=2,\ c=\sec^2u$$  

Multiple Choice

Given is first few steps of Chain Rule differentiation of the function. What is $$p?$$  

• $$\frac{2}{3}\operatorname{cosec}\ u$$  

• $$\operatorname{cosec}\ u$$  

• $$\frac{2}{3}\operatorname{cosec}\ \left(1-2x\right)$$

Multiple Choice

Given is the continuation steps of previous  Chain Rule differentiation of the function. What is $$q?$$  

• $$-\frac{4}{3}\operatorname{cosec}u\ \cot\ u$$

• $$-\frac{4}{3}\operatorname{cosec}\left(1-2x\right)\cot\ \left(1-2x\right)$$

• $$\frac{2}{3}\operatorname{cosec}\ \left(1-2x\right)$$

• $$\frac{4}{3}\operatorname{cosec}\left(1-2x\right)\cot\ \left(1-2x\right)$$

​ARE YOU CONFIDENT DOING THE QUESTIONS INDEPENDENTLY?

​LEVEL 3

​DOING PIT-STOP 2 (CHAIN RULE)

​

​INDEPENDENTLY

​ARE YOU SURE

​ YOU ARE READY?

​LET'S DO THIS CHAMPS......

Multiple Choice

Given  $$y=\sin5x$$  .Find $$\frac{\text{d}y}{\text{d}x}$$  

• $$\cos5x$$  

• $$-\cos5x$$  

• $$5\cos5x$$  

• $$-5\cos5x$$  

Multiple Choice

Given  $$y=\frac{1}{5}\sin5x$$  .Find $$\frac{\text{d}y}{\text{d}x}$$  

• $$\cos5x$$  

• $$\frac{1}{5}\cos5x$$  

• $$-\frac{1}{5}\cos5x$$

• $$-\cos5x$$  

Multiple Choice

Given  $$y=\cos2x$$  .Find $$\frac{\text{d}y}{\text{d}x}$$  

• $$\sin2x$$  

• $$-\sin2x$$  

• $$2\sin2x$$  

• $$-2\sin2x$$  

Multiple Choice

Given  $$y=\cot6x$$  .Find $$\frac{\text{d}y}{\text{d}x}$$  

• $$\operatorname{cosec}^26x$$  

• $$-\operatorname{cosec}^26x$$  

• $$-6\operatorname{cosec}^26x$$  

• $$6\operatorname{cosec}^26x$$  

Multiple Choice

Given  $$y=\frac{1}{6}\cot6x$$  .Find $$\frac{\text{d}y}{\text{d}x}$$  

• $$\operatorname{cosec}^26x$$  

• $$-\operatorname{cosec}^26x$$  

• $$-6\operatorname{cosec}^26x$$  

• $$6\operatorname{cosec}^26x$$  

Multiple Choice

Given  $$y=\frac{1}{6}\tan\left(1-6x\right)$$  .Find $$\frac{\text{d}y}{\text{d}x}$$  

• $$\sec^2\left(1-6x\right)$$  

• $$-\sec^2\left(1-6x\right)$$  

• $$\frac{1}{6}\sec^2\left(1-6x\right)$$  

• $$-\frac{1}{6}\sec^2\left(1-6x\right)$$

Multiple Choice

Given  $$y=-\frac{1}{2}\cos2x$$  .Find $$\frac{\text{d}y}{\text{d}x}$$  

• $$\sin2x$$  

• $$-\frac{1}{2}\sin2x$$  

• $$\frac{1}{2}\sin2x$$  

• $$-\sin2x$$  

Multiple Choice

Given  $$y=\frac{1}{5}\sin\left(1-5x\right)$$  .Find $$\frac{\text{d}y}{\text{d}x}$$  

• $$\frac{1}{5}\cos\left(1-5x\right)$$  

• $$-\frac{1}{5}\cos\left(1-5x\right)$$  

• $$-\cos\left(1-5x\right)$$  

• $$\cos\left(1-5x\right)$$  

Multiple Choice

Given  $$y=\frac{2}{5}\sin\left(5x-1\right)$$  .Find  $$\frac{\text{d}y}{\text{d}x}$$  

• $$2\cos\left(5x-1\right)$$  

• $$\frac{2}{5}\cos\left(5x-1\right)$$  

• $$-\frac{2}{5}\cos\left(5x-1\right)$$  

• $$-2\cos\left(5x-1\right)$$  

Multiple Choice

Given $$y=\frac{1}{2}\cos\left(1-2x\right)$$  , find $$\frac{\text{d}y}{\text{d}x}$$  

• $$\frac{1}{2}\sin\left(1-2x\right)$$  

• $$-\frac{1}{2}\sin\left(1-2x\right)$$  

• $$\sin\left(1-2x\right)$$  

• $$-\sin\left(1-2x\right)$$  

Multiple Choice

Given $$y=-\frac{1}{2}\cos\left(2x-1\right)$$  , find $$\frac{\text{d}y}{\text{d}x}$$  

• $$\frac{1}{2}\sin\left(2x-1\right)$$  

• $$-\frac{1}{2}\sin\left(2x-1\right)$$  

• $$\sin\left(2x-1\right)$$  

• $$-\sin\left(2x-1\right)$$  

Multiple Choice

Given $$y=-\frac{1}{6}\cot6x$$  , find $$\frac{\text{d}y}{\text{d}x}$$  

• $$\operatorname{cosec}^26x$$  

• $$-\frac{1}{6}\operatorname{cosec}^26x$$  

• $$\frac{1}{6}\operatorname{cosec}^26x$$  

• $$-\operatorname{cosec}^26x$$  

Multiple Choice

Given  $$y=\frac{1}{6}\cot\left(1-6x\right)$$  

• $$\frac{1}{6}\operatorname{cosec}^2\left(1-6x\right)$$  

• $$-\operatorname{cosec}^2\left(1-6x\right)$$  

• $$\operatorname{cosec}^2\left(1-6x\right)$$  

• $$-\frac{1}{6}\operatorname{cosec}^2\left(1-6x\right)$$  

Multiple Choice

Given $$y=\sec7x$$  ,find $$\frac{\text{d}y}{\text{d}x}$$  

• $$\sec7x\tan7x$$  

• $$7\sec7x\tan7x$$  

• $$-7\sec7x\tan7x$$

• $$\frac{1}{7}\sec7x\tan7x$$

​TIRED??

Multiple Choice

Given  $$y=\frac{1}{7}\sec7x$$  

• $$\sec7x\tan7x$$  

• $$7\sec7x\tan7x$$  

• $$-\sec7x\tan7x$$  

• $$-7\sec7x\tan7x$$  

Multiple Choice

Given $$y=-\frac{1}{7}\sec7x$$  

• $$-\sec7x\tan7x$$  

• $$\sec7x\tan7x$$  

• $$\frac{1}{7}\sec7x\tan7x$$  

• $$-7\sec7x\tan7x$$  

Multiple Choice

Given  $$y=\frac{1}{7}\sec\left(1-7x\right)$$  ,find $$\frac{dy}{dx}$$  

• $$\sec\left(1-7x\right)\tan\left(1-7x\right)$$  

• $$-\sec\left(1-7x\right)\tan\left(1-7x\right)$$  

• $$\frac{1}{7}\sec\left(1-7x\right)\tan\left(1-7x\right)$$

• $$-\frac{1}{7}\sec\left(1-7x\right)\tan\left(1-7x\right)$$

Multiple Choice

Given  $$y=-\frac{1}{7}\sec\left(1-7x\right)$$  ,find $$\frac{dy}{dx}$$  

• $$\sec\left(1-7x\right)\tan\left(1-7x\right)$$  

• $$-\sec\left(1-7x\right)\tan\left(1-7x\right)$$  

• $$\frac{1}{7}\sec\left(1-7x\right)\tan\left(1-7x\right)$$

• $$-\frac{1}{7}\sec\left(1-7x\right)\tan\left(1-7x\right)$$

Multiple Choice

Given  $$y=\operatorname{cosec}\ 8x$$  , find  $$\frac{dy}{dx}$$  

• $$-\operatorname{cosec}8x\cot8x$$  

• $$8\operatorname{cosec}8x\cot8x$$  

• $$-8\operatorname{cosec}8x\cot8x$$  

• $$\operatorname{cosec}8x\cot8x$$  

Multiple Choice

Given  $$y=\frac{1}{8}\operatorname{cosec}\ 8x$$  , find  $$\frac{dy}{dx}$$  

• $$-\operatorname{cosec}8x\cot8x$$  

• $$8\operatorname{cosec}8x\cot8x$$  

• $$-8\operatorname{cosec}8x\cot8x$$  

• $$\operatorname{cosec}8x\cot8x$$  

Multiple Choice

Given  $$y=-\frac{1}{8}\operatorname{cosec}\ 8x$$  , find  $$\frac{dy}{dx}$$  

• $$-\operatorname{cosec}8x\cot8x$$  

• $$8\operatorname{cosec}8x\cot8x$$  

• $$-8\operatorname{cosec}8x\cot8x$$  

• $$\operatorname{cosec}8x\cot8x$$  

Multiple Choice

Given  $$y=-\frac{1}{8}\operatorname{cosec}\ \left(8x-1\right)$$  , find  $$\frac{dy}{dx}$$  

• $$-\operatorname{cosec}\left(8x-1\right)\cot\left(8x-1\right)$$  

• $$8\operatorname{cosec}\left(8x-1\right)\cot\left(8x-1\right)$$  

• $$-8\operatorname{cosec}\left(8x-1\right)\cot\left(8x-1\right)$$  

• $$\operatorname{cosec}\left(8x-1\right)\cot\left(8x-1\right)$$  

Multiple Choice

Given  $$y=\frac{1}{8}\operatorname{cosec}\ \left(1-8x\right)$$  , find  $$\frac{dy}{dx}$$  

• $$-\operatorname{cosec}\left(1-8x\right)\cot\left(1-8x\right)$$  

• $$8\operatorname{cosec}\left(1-8x\right)\cot\left(1-8x\right)$$  

• $$-8\operatorname{cosec}\left(1-8x\right)\cot\left(1-8x\right)$$  

• $$\operatorname{cosec}\left(1-8x\right)\cot\left(1-8x\right)$$  

Multiple Choice

Given $$y=\tan3x$$  ,find $$\frac{dy}{dx}$$  

• $$\sec^23x$$  

• $$3\sec^23x$$  

• $$\sec^2x$$  

• $$3\sec^2x$$  

Multiple Choice

Given $$y=\frac{1}{3}\tan3x$$  ,find $$\frac{dy}{dx}$$  

• $$\sec^23x$$  

• $$3\sec^23x$$  

• $$\sec^2x$$  

• $$3\sec^2x$$  

Multiple Choice

Given $$y=-\frac{1}{3}\tan3x$$  ,find $$\frac{dy}{dx}$$  

• $$\sec^23x$$  

• $$3\sec^23x$$  

• $$-\sec^23x$$  

• $$-3\sec^23x$$  

Multiple Choice

Given $$y=-\frac{1}{3}\tan\left(1-3x\right)$$  ,find $$\frac{dy}{dx}$$  

• $$\sec^2\left(1-3x\right)$$  

• $$3\sec^2\left(1-3x\right)$$  

• $$-\frac{1}{3}\sec^2\left(1-3x\right)$$  

• $$-3\sec^2\left(1-3x\right)$$  

Multiple Choice

Given $$y=-\frac{1}{3}\tan\left(3x-1\right)$$  ,find $$\frac{dy}{dx}$$  

• $$\sec^2\left(1-3x\right)$$  

• $$-\sec^2\left(1-3x\right)$$  

• $$-\frac{1}{3}\sec^2\left(1-3x\right)$$  

• $$-3\sec^2\left(1-3x\right)$$  

​WHAT YOU SHOULD HAVE UNLOCKED HERE?

​

​1. able to identify which needs PIT-STOP 2 (Chain rule)

​2. How to carry out the process independently

​3. Please review the questions that you are unsure before trying the next subsection