The chain rule part 2

Match

Warm Up

Match the following

• $$\frac{d}{dx}\left(200\right)=$$  

• $$\frac{d}{dx}\left(200x\right)=$$  

• $$\frac{d}{dx}\left(x^{200}\right)=$$  

• $$\frac{d}{dx}\left(\frac{1}{200}x^{200}\right)=$$  

• $$\frac{d}{dx}\left(200x^2+200x\right)=$$  

0

200

200x¹⁹⁹

x¹⁹⁹

400x+200

• 0

• 200

• 200x¹⁹⁹

• x¹⁹⁹

• 400x+200

Objectives 1.Recall the general power rule and chain rule.
2. Apply the chain rule to real-life UAE-based models (e.g., population, temperature functions).

Match

Match the following

• $$\frac{d}{dx}\left(20\right)=$$  

• $$\frac{d}{dx}\left(20x\right)=$$  

• $$\frac{d}{dx}\left(x^{20}\right)=$$  

• $$\frac{d}{dx}\left(\frac{1}{20}x^{20}\right)=$$  

• $$\frac{d}{dx}\left(20x^2+20x\right)=$$  

0

20

20x¹⁹

x¹⁹

40x+20

• 0

• 20

• 20x¹⁹

• x¹⁹

• 40x+20

Match

Match the following

• $$\frac{d}{dx}\left(\cos x\right)$$

• $$\frac{d}{dx}\left(\tan x\right)$$

• $$\frac{d}{dx}\left(\sec x\right)$$

• $$\frac{d}{dx}\left(\sin x\right)$$

• $$\frac{d}{dx}\left(\cot x\right)$$

- sin x

sec²x

secx tanx

cos x

-csc²x

• - sin x

• sec²x

• secx tanx

• cos x

• -csc²x

Multiple Choice

$$find\ g'\left(x\right)\ for\ g\left(x\right)=\left(h\left(x\right)\right)^3$$  

• $$g'\left(x\right)=3\left(h'\left(x\right)\right)^2$$  

• $$g'\left(x\right)=3h'\left(x\right)^2$$  

• $$g'\left(x\right)=3\left(h\left(x\right)\right)^2\cdot h'\left(x\right)$$  

• $$g'\left(x\right)=3h\left(x\right)^2\cdot h'\left(x\right)$$  

Multiple Choice

Find $$\frac{dr}{d\theta}\ if\ r=\left(\sec\theta+\tan\theta\right)^{-3}$$  

• $$-3\left(\sec\theta+\tan\theta\right)^{-4}\cdot\sec\theta$$  

• $$-3\left(\sec\theta+\tan\theta\right)^{-4}\left(\sec\theta\tan\theta+\sec^2\theta\right)$$  

• $$-3\left(\sec\theta+\tan\theta\right)^{-4}$$  

• $$-3\left(\sec\theta\tan\theta+\sec^2\theta\right)^{-4}$$  

Multiple Choice

$$\frac{\text{d}}{\text{d}x}\tan^2\left(3x\right)$$  

• $$2\tan\left(3x\right)\sec^2\left(x\right)\cdot3$$  

• $$2\tan\left(3x\right)\sec^2\left(3x\right)\cdot3$$  

• $$2\sec^2\left(3x\right)$$  

• $$2\sec^2\left(3x\right)\cdot3$$  

Multiple Choice

Given the function $$y=\sqrt{5x-7}$$  find  $$\frac{\text{d}x}{\text{d}y}$$ . 

• $$\frac{5}{2\sqrt{5x-7}}$$  

• $$\frac{1}{2\sqrt{5x-7}}$$  

• $$\frac{5}{\sqrt{5x-7}}$$  

• $$5\sqrt{5x-7}$$  

Multiple Choice

$$\frac{dy}{dt}\ if\ y=\cos\sqrt{10t+12}$$  Find

• $$\frac{-5}{\sqrt{10t+12}}\sin\sqrt{10t+12}$$  

• $$\frac{-1}{2\sqrt{10t+12}}\sin\sqrt{10t+12}$$  

• $$-\sin\sqrt{10t+12}$$  

• $$-\sin\left(\frac{5}{\sqrt{10t+12}}\right)$$