Derivative Rules

Derivative Rules

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Multiple Choice

What is the derivative of y=4

• 1

• 4

• 0

Multiple Choice

Find the derivative of:

$$f\left(x\right)=4\sqrt{x}$$  

• $$f'\left(x\right)=\frac{2}{\sqrt{x}}$$  

• $$f'\left(x\right)=2\sqrt{x}$$  

• $$f'\left(x\right)=4x^{\frac{1}{2}}$$  

• $$f'\left(x\right)=2x^{-\frac{1}{2}}$$  

Multiple Choice

Find the derivative of:

$$f\left(x\right)=x^9-7x^5+3x+13$$  

• $$f'\left(x\right)=9x^8-35x^4+3$$  

• $$f'\left(x\right)=x^8-7x^4+3$$  

• $$f'\left(x\right)=9x^9-35x^5+3x$$  

• $$f'\left(x\right)=9x^8-35x^4+3+13$$  

​Let's Add Some more

Hint: Write these down!

Multiple Choice

$$\frac{d}{dx}\left[\ln x\right]=$$  

• $$e^x$$  
  

• $$\frac{1}{x}$$  
  

• $$\frac{1}{\ln x}$$  
  

• $$\frac{1}{e^x}$$  
  

Multiple Choice

$$\frac{d}{dx}\left[e^x\right]=$$  

• $$xe^{\left(x-1\right)}$$  

• $$e^x$$  

• $$xe^x\left(x-1\right)$$  
  

• $$\ln x$$  

Multiple Choice

Derivative of tanx

• cosx

• sinx

• secx

• sec²x

Multiple Choice

What is the derivative of cos(x)?

• sin(x)
• cos(x)
• -sin(x)
• -cos(x)

Multiple Choice

What is the derivative of y = 2π? *Tricky, what is Pi? Think hard....

• 0

• 2

• -2

• None of the above

Multiple Choice

Let f(x) = (x - 1)(x + 2). Find f'(0)

• 0

• 1

• 2

• 3

Multiple Choice

Use the product rule to find the derivative. $$f\left(x\right)\ =\ -x^3\left(3x^4-2\right)$$  

• $$-3x^2+12x^3$$  

• $$-21x^3+6x$$  

• $$-21x^6+6x^2$$  

• $$-84x^6-6x^2$$  

​Write this down!

Multiple Choice

Which option is using correctly the quotient rule?

Multiple Choice

• f'(x)=(2x⁵ – 10)/ (x⁵–5)²

• f'(x)=(10x⁴)/ (x⁵–5)²

• f'(x)=2(x⁵– 5)/ (x⁵–5)²

• f'(x)=(2x⁵)/ (x⁵–5)

Multiple Choice

Find the derivative

• 8x⁴ -60x²
• 16x⁴
• (8x⁴ -60x²)/(2x²-5)²
• 3x

Multiple Choice

$$y=\frac{x^2+6}{2x−7}.$$  

• $$\frac{2x^2−14x−12}{(2x−7)^2}$$  

• $$\frac{4x^2−12x−2}{(2x−7)^2}$$  

• $$\frac{x^2−x+12}{(2x−7)^2}$$  

• $$\frac{2x^2+14x+12}{(2x−7)^2}$$  

Multiple Choice

Find the derivative.

$$f\left(x\right)=\frac{\tan x}{2x-3}$$

• $$f'\left(x\right)=\frac{\left(2x-3\right)\sec x\tan x-2\tan x}{\left(2x-3\right)^2}$$

• $$f'\left(x\right)=\frac{\left(2x-3\right)\sec^2x-2\tan x}{\left(2x-3\right)^2}$$

• $$f'\left(x\right)=\frac{\sec^2x-2\tan x}{\left(2x-3\right)^2}$$

• $$f'\left(x\right)=\frac{\left(2x-3\right)\csc^2x-2\tan x}{\left(2x-3\right)^2}$$

Let's Review

Multiple Choice

Find the derivative of y = 2x² (3x - 4)

• 6x³ - 8x²
• 18x² - 16x
• 15x² - 6x
• 6x - 4

Multiple Choice

Find the derivative
f(t) = (t² + 2t)⁵

• f'(t) = 5(2t+2)⁴
• f'(t) = 5(t² + 2t)⁴
•  f'(t) = 5(t² + 2t)⁴(2t)
• f'(t) = 5(t² + 2t)⁴(2t + 2)

Multiple Choice

• 8x⁴ -60x²
• 16x⁴
• (8x⁴ -60x²)/(2x²-5)²
• 3x

Multiple Choice

Which if the following is not a correct notation for 'derivative?'

• f'(x)
• y'
• dy
• dy/dx

Multiple Choice

Derivative means the same thing as

• slope of the tangent line
• slope of the normal line
• exponent
• slope of the secant line

Multiple Choice

Which rule would you need to use?

• Power Rule
• Product Rule
• Quotient Rule
• Chain Rule

Multiple Choice

Differentiate (2x + 2)(x - 5)⁸ with respect to x.

• (2x + 2)(x - 5)⁷ + 2(x - 5)⁸
• 8(2x + 2)(x - 5)⁷ + 2(x - 5)⁸
• 2(x - 5)⁸
• 16(x - 5)⁷

Multiple Choice

Find the derivative.
f(x) = -8x^-3 + 5x - e^x

• f'(x) = -24x^-4 + 5 - e^x
• f'(x) = 24x^-4 + 5 - e^x
• f'(x) = 24x^-2 + 5 - e^x
• f'(x) = 24x^-4 + 5 - e^x-1

Multiple Choice

The derivative of 

• y'=2x-x^-2
• y'=x^-1+8x
• y'=x^-2+8x
• y=8x-x^-2

Multiple Choice

Find the derivative:
y=log₉x

• y′=1 /((log9)x)
• y′=(ln9)/x
• y′=1/(x(lnx))
• y′=1/((ln9)x)

Multiple Choice

Differentiate      y= 12x^-2

• 24x^-1
• -24x^-3
• -24x^-1
• 6x^-3