Derivatives of Trigonometric Functions 24

​Derivatives of Trigonometric Functions

Match

Warm Up

Match the following

• $$\frac{d}{dx}\left(\cos2x\right)$$

• $$\frac{d}{dx}\left(\tan3x\right)$$

• $$\frac{d}{dx}\left(\sec8x\right)$$

• $$\frac{d}{dx}\left(\sin x\right)$$

• $$\frac{d}{dx}\left(\cot x\right)$$

- 2sin 2x

3sec² 3x

8sec8x tan8x

cos x

-csc²x

• - 2sin 2x

• 3sec² 3x

• 8sec8x tan8x

• cos x

• -csc²x

Multiple Choice

BEFORE

The Dubai Metro train runs according to the function

$$f\left(x\right)=\sqrt[]{\sin4x}$$

Find the derivative of the function?

• $$\frac{2\cos4x}{\sqrt[]{\sin4x}}$$

• $$\frac{2\cos x}{\sqrt[]{\sin x}}$$

• $$\frac{\cos4x}{2\sqrt[]{\sin4x}}$$

• $$\frac{4\cos4x}{\sqrt[]{\sin4x}}$$

​Learning objectives

​•Find the derivatives of trigonometric functions using differentiation rules.

•Solve physical and engineering problems using derivatives of trigonometric functions.

Multiple Choice

Emsat

Find $$\frac{\text{d}}{\text{d}x}\left(\ 4\sin\ 6x\right)=$$  

• 24 cos x

• 24 sin 6x

• 24 cos 6x

• -24 cos 6x

Multiple Choice

If $$y\ =\ 3x\ -\ 5\ \cos\left(2x^2-3\right)$$  . Find $$\frac{\text{d}y}{\text{d}x}$$  .

• $$3+\ 20x\ \sin\ \left(2x^2-3\right)$$  

• $$3-20x\ \sin\left(2x^2-3\right)$$   

• $$3\ -\ 20\ \sin\ \left(2x^2-3\right)$$  

• $$3\ +\ 20\ \sin\left(2x^2-3\right)$$  

Multiple Choice

Derive $$y\ =\ 4x\ \tan\ 6x.$$  

• $$4x\ \tan\ 6x\ +\ 24\ \sec^26x$$  

• $$4x\ \tan\ 6x\ +\ 24x\ \csc^26x$$  

• $$4\ \tan\ 6x\ +\ 24\ x\ \sec^26x$$  

• $$4\tan6x\ -\ 24x\ \csc^26x$$  

Multiple Choice

Find $$\frac{\text{d}}{\text{d}x}\left(\cot\ ^34x\right).$$  

• $$12\ \cot^24x\ \csc^24x$$  

• $$-12\cot^34x\ \csc^24x$$  

• $$-12\ \cot^24x\ \csc^24x$$  

• $$-12\ \cot^24x\ \csc\ 4x$$  

Multiple Choice

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

• 1

• $$-\sin x\cos x$$  

• $$\cos^2x-\sin^2x$$  

Multiple Choice

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

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

• $$2\cos\left(2x\right)$$  

• $$4\cos\left(2x\right)$$  

​Equation of Tangent Line.

Multiple Choice

Derivative means the same thing as

• slope of the tangent line

• slope of the normal line

• slope of the secant line

• the function’s value

Open Ended

Find an equation of the tangent line

to $$y=\sin\ 4x\ \ at\ a=\frac{\pi}{8}$$

Multiple Choice

Find an equation of a tangent line to

$$f\left(x\right)=3\cos x\ +x\ at\ x=\frac{\pi}{2}$$

• $$y=-2\left(x-\frac{\pi}{2}\right)+\frac{\pi}{2}$$

• $$y=2\left(x-\frac{\pi}{2}\right)+\frac{\pi}{2}$$

• $$y=-2\left(x+\frac{\pi}{2}\right)+\frac{\pi}{2}$$

• $$y=-2\left(x-\frac{\pi}{4}\right)+\frac{\pi}{4}$$

Math Response

Your turn

Find an equation of the tangent line to

$$y=\tan\ 3x\ at\ a=0$$ .

Type answer here

Correct Answer: y=3x

Deg^°

Rad

Multiple Choice

Use the position function to find the Velocity at time 𝒕 = 𝒕𝟎.

$$f\left(x\right)=x\sin x\ at\ x=\frac{\pi}{2}$$

• $$1$$

• $$2$$

• $$3$$

• $$4$$