Calculus Derivatives

Limits and Derivatives Relationship

Limits

• Limits measure what a function looks like its doing.

• Follow the path on the graph. We do not care what happens at the actual point.

Limits

Multiple Choice

The slope of a function is described by its ____________.

• first derivative

• second derivative

• third derivative

• expression

Multiple Choice

Find the derivative of the given equation $$f\left(x\right)=4x$$  

• 4

• x

• -4

• 0

Multiple Choice

What is the derivative of the function

$$y=6x^3$$  

• $$\frac{dy}{dx}=18x^2$$  

• $$\frac{dy}{dx}=6x^2$$

• $$\frac{dy}{dx}=18x$$  

• 6 

Multiple Choice

What is the derivative?

$$y=6x^3-3x^2+x-7\$$  

• $$\frac{\text{d}y}{\text{d}x}=18x^2-6x+1$$  

• $$\frac{\text{d}y}{\text{d}x}=18x^3-6x-7$$  

• $$\frac{\text{d}y}{\text{d}x}=6x^2-3x+1$$  

• $$\frac{\text{d}y}{\text{d}x}=18x^2-6x+1-7$$  

Multiple Choice

What is the derivative of $$f\left(x\right)=7$$  ?

• 0

• 1

• x

• 7x

Multiple Choice

If $$y=\sqrt{x}$$  then $$\frac{dy}{dx}=...$$  

• $$\frac{1}{2\sqrt{x}}$$  

• 0

• 1

• $$\frac{1}{x^2}$$  

Multiple Choice

Differentiate:

$$6x^{\frac{2}{3}}$$

• $$4x^{-\frac{1}{3}}$$

• $$6x^{-\frac{2}{3}}$$

• $$6x^{-\frac{1}{3}}$$

• $$4x^{\frac{1}{3}}$$

Multiple Choice

Find the derivative of the given equation
$$f\left(x\right)=x-x^2$$

• 1 - 2x

• -2x

• -2

• 2x

Multiple Choice

Differentiate $$y=3x^2+5x-3$$  

• $$\frac{dy}{dx}=6x+5$$  

• $$\frac{dy}{dx}=x^3+\frac{5}{2}x^2-3x$$  

• $$\frac{dy}{dx}=3x+5$$  

• $$\frac{dy}{dx}=x-3$$  

Multiple Choice

Given $$f\left(x\right)=2x^3-4x^2+7x-5$$ ,

find  $$f'\left(x\right)$$ . 

• $$6x^2-8x+7-5$$  

• $$6x^2-8x+7$$  

• $$6x^3-8x^2+7x-5$$  

• $$2x^2-4x+7$$  

Multiple Choice

Find the derivative of:
f(x) = 7(x + 4)⁵

• 35(x + 4)⁵

• 35(x + 4)⁴

• 140(x + 4)⁵

• 140(x + 4)⁴

Multiple Choice

If $$g\left(x\right)=\left(2x-5\right)^6$$  , then $$g'\left(x\right)=$$  ...

• $$10\left(2x-5\right)^5$$  

• $$12\left(2x-5\right)^5$$  

• $$6\left(2x-5\right)^5$$  

• $$24\left(2x-5\right)^5$$  

Multiple Choice

First derivative of $$f\left(x\right)=\left(2x^2-3x+1\right)^5$$  is ...

• $$5x\left(2x^2-3x+1\right)^4$$  

• $$\left(4x-3\right)\left(2x^2-3x+1\right)^4$$  

• $$\left(16x-3\right)\left(2x^2-3x+1\right)^4$$   

• $$\left(20x-15\right)\left(2x^2-3x+1\right)^4$$  

Multiple Choice

Find the derivative of f(x)=(x³-2x)²

• 6x⁵ - 12x³+8x

• 6x⁵ - 16x³+8x

• x⁶-4x⁴+4x²

• 6x⁵ - 16x^3-8x

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

Multiple Choice

At which point(s) will the slope(s) of the tangent line be POSITIVE?

• at C only

• at points A, C and E only

• at point B and D only

• at points A and E only

Multiple Choice

At which point(s) will the slopes of the tangent line be zero?

• at C only

• at points A, C and E only

• at point B and D only

• at points A and E only

Multiple Choice

$$f\left(x\right)=\ x^2+2x\ -3\$$  The slope of a tangent line to the graph of the given function at (1, 0) is 

• 4

• 1

• 2

• 3

Multiple Choice

Find $$f'\left(2\right)$$  if $$f\left(x\right)=\left(x^2-3\right)^4$$  

• 1

• 4

• 8

• 16

Multiple Select

Which of the following are synonyms for find the derivative?

• f'(x)

• dy/dx

• y'

• slope of the tangent line

• instantaneous rate of change

Multiple Choice

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

• f'(x)

• y'

• dy

• dy/dx

Multiple Choice

If a function has a derivative that is negative, what does that tell you?

• The function is increasing

• The function is decreasing

Multiple Choice

How many derivatives can you take of a function?

• only 3

• only 2

• only 1

• until you can no longer derive it