Calculus: Derivatives

Question 1

What is an instantaneous rate of change?

Answer 1

The rate of change at a specific point on a function

Question 2

What is a derivative?

Answer 2

The derivative of a function at a given point represents the instantaneous rate of change of the function at that point. In other words, it tells you how fast the function is changing at that specific moment.

Question 3

What is the relationship between a tangent line and a derivative?

Answer 3

The gradient (a.k.a. slope) of a tangent line to a specific point is equal to the derivative at that point.

Question 4

What is a tangent line?

Answer 4

A line which touches (but does not cross through) a curve at a particular point. The gradient of the tangent line is equal to the derivative of the curve at that point.

Question 5

Where is the derivative of a curve equal to zero?

Answer 5

At the local max, local min, or inflection point

Question 6

How to find the derivative of a function?

Answer 6

Use the power rule

Question 7

What is the power rule?

Answer 7

1) Multiply the coefficient by the exponent
2) Subtract 1 from the exponent

Question 8

What is the derivative of a constant term?

Answer 8

0 (when there is a constant term, it just disappears)

Question 9

What is the derivative of function f(x)?

Answer 9

f’(x)=15x^4

Explanation:

Question 10

Find the derivative

Answer 10

Question 11

What does “differentiate” mean?

Answer 11

Find the derivative

Question 12

What notation do we use to represent the derivative?

Answer 12

f’(x) “f prime of x”

dy/dx “dee why dee ex”

Question 13

Differentiate the function

Answer 13

Question 14

Differentiate the equation

Answer 14

Question 15

Differentiate y with respect to x

Answer 15

Question 16

How do you find the equation of a tangent line at a particular point?

Answer 16

1) Find the derivative using the power rule.
2) Plug in the x-value to the derivative equation to find the gradient of the tangent line.
3) Use point-gradient form of a line to find the equation.

Question 17

What is a normal line?

Answer 17

a line perpendicular to the tangent line

Question 18

What does this mean?

Answer 18

It means derivative. It’s just another way of writing

f’(x).

Question 19

What should you do if you are asked to

“find the gradient of the graph of f at x=3 “?

Answer 19

• By hand (and at least written on exam for method marks):

1. Take the derivative of f.
2. Plug 3 into this function. Make sure to write down “ f’(3) “.

• On the calculator (to double check)

Question 20

How can you use the derivative to find out on which intervals the original function is increasing/decreasing?

Answer 20

• When the derivative is negative (f’(x)<0), the function is decreasing.
• When the derivative is positive (f’(x)>0), the function is increasing.

Question 21

What should you do when asked to “minimize”, “maximize”, or “optimize” a function?

Answer 21

• Option 1:
  Graph the function on your GDC and use it to find the min/max
• Option 2:

1. Find the derivative of the function, f’(x).
2. Set the derivative to zero, f’(x)=0.
3. Solve for x.
4. Plug in x from the previous step to find the y-value.

Question 22

How are derivatives related to the max/min of a function?

Answer 22

The tangent at the min or max of a function is a flat line, which means its slope is 0. Therefore, the min/max occurs at f’(x)=0.

You find the derivative and then solve for x when f’(x)=0. (This gives you the x-coordinate. To find the y-coordinate, you have to plug this x into the original function.)