Trigonometric Derivatives

Trigonometric Derivatives

Multiple Choice

Warm up: What is 

 $\sin\left(\frac{3\pi}{2}\right)$  

• 1

• -1

• 0

• Undefined

Today we will learn...

• How to differentiate trigonometric functions

It's important to remember with trigonmetric derivatives that all differentiation rules remain in place

This includes product rule, chain rule, quotient rule, etc.

 $\frac{d}{dx}\left[\sin x\right]=\cos x$  
With chain rule, 
 $\frac{d}{dx}\left[\sin\left(f\left(x\right)\right)\right]=\cos\left(f\left(x\right)\right)f'\left(x\right)$  

Multiple Choice

 $\frac{d}{dx}\left[\sin\left(2x\right)\right]$  

•  $\cos\left(2x\right)$  

•  $2\sin\left(2x\right)$  

•  $2\cos\left(2x\right)$  

Multiple Choice

 $\frac{d}{dx}\left[\sin\left(x^2+x\right)\right]$  

•  $\left(2x+1\right)\cos\left(x^2+x\right)$  

•  $\cos\left(2x+1\right)$  

•  $\cos\left(x^2+x\right)$  

 $\frac{d}{dx}\left[\cos x\right]=-\sin x$  

With chain rule, 

 $\frac{d}{dx}\left[\cos\left(f\left(x\right)\right)\right]=-\sin\left(f\left(x\right)\right)f'\left(x\right)$  

Multiple Choice

 $\frac{d}{dx}\left[\cos\left(e^x\right)\right]$  

•  $-\sin\left(e^x\right)$  

•  $e^x$  

•  $-e^x\sin\left(e^x\right)$  

Multiple Choice

 $\frac{d}{dx}\left[\sin x\cos x\right]$  

• 1

•  $-\sin x\cos x$  

•  $\cos^2x-\sin^2x$  

Multiple Choice

 $\frac{d}{dx}\left[\sin^{2\ }\left(2x\right)\right]$  

•  $4\sin\left(2x\right)\cos\left(2x\right)$  

•  $2\cos\left(2x\right)$  

•  $4\cos\left(2x\right)$  

Rewrite  $\tan x$  using  a trig identity and try to find its derivative using quotient rule

 $\frac{d}{dx}\left[\tan x\right]=\sec^2x$  

With Chain Rule

 $\frac{d}{dx}\left[\tan\left(f\left(x\right)\right)\right]=\sec^2\left(f\left(x\right)\right)f'\left(x\right)$  

Rewrite  $\cot x$  using  a trig identity and try to find its derivative using quotient rule

 $\frac{d}{dx}\left[\cot x\right]=-\csc^2x$  

With Chain Rule

 $\frac{d}{dx}\left[\cot\left(f\left(x\right)\right)\right]=-\csc^2\left(f\left(x\right)\right)f'\left(x\right)$  

Multiple Choice

Find the derivative f(x) = tanxcosx

• f'(x) = sec²xcosx - tanxsinx
• f'(x) = sec²xcosx + tanxsinx
• f'(x) = sec²xsinx
• f'(x) = sec²xcosx - tanxcosx

Multiple Choice

Determine the derivative of:  f(x) = x⁴sinx

• x⁴ cosx - 4x³sinx
• x⁴ cosx + 4x³sinx
• 4x³cosx
• -4x³cosx

Multiple Choice

 $\frac{d}{dx}\left[\cos^{2\ }\left(\sqrt{x}\right)\right]$  

•  $2\cos\sqrt{x}\sin\sqrt{x}$  

•  $-\frac{\cos\sqrt{x}\sin\sqrt{x}}{\sqrt{x}}$  

•  $\frac{\cos^{2\ }\left(\sqrt{x}\right)}{2\sqrt{x}}$