Derivatives Theory SL

Question 1

What is the mathematical definition of derivative?

Answer 1

Question 2

How are derivatives and tangent lines related?

Answer 2

The derivative is the slope of the tangent line for a given x.

Question 3

What is the power rule for finding derivatives?

Answer 3

Question 4

What is the derivative of a constant?

Answer 4

0.

When you take the derivative of a constant, that term just disappears.

Question 5

How do you find the tangent line of a function at a given point (x₁,y₁)?

Answer 5

Take the derivative of the function to find f’(x), giving you the slope of the tangent line. Since the tangent line is straight, it can be represented as y=mx+c.

Plug in x₁ and y₁ and solve for c.

y₁=f’(x₁)*x₁+c

Question 6

How do you take the derivative of a function with two or more terms?

For example, f(x)=4x³+2x²-3

Answer 6

You apply the sum or difference rule. The derivative of a function that adds or subtracts two or more terms is the derivative of each term added or subtracted together. You just take the derivative of each term.

For example, f’(x)=8x²+4x.

Question 7

What is the derivative of e^x?

Answer 7

e^x

Question 8

What is the derivative of ln(x)?

Answer 8

Question 9

What is the product rule and when do you need it?

Answer 9

You need the product rule when your function is actually two simpler functions multiplied together.

Question 10

What is the quotient rule and when do you need it?

Answer 10

You need the quotient rule when your function is actually two simpler functions divided.

Question 11

When do you need to use the chain rule?

Answer 11

When you have functions inside of other functions. For example:

Question 12

What is the chain rule?

Answer 12

Question 13

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

Answer 13

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.)

Question 14

How are position, velocity, and acceleration related?

Answer 14

• Position just tells you where a point is or how far something has travelled.
• Velocity is the derivative of position.
• Acceleration is the derivative of velocity.

Question 15

How can you use the derivative to graph a sketch of a function, f(x)?

Answer 15

1. Find f’(x)=0 to see where the local min/max are.
2. Find when f’(x)<0 to see where the function is decreasing.
3. Find when f’(x)>0 to see where the function is increasing.
4. Find f’‘(x)=0 to see where the inflection points are.
5. Find f’‘(x)<0 to see where the function is concave down (frowny face).
6. Find f’‘(x)>0 to see where the function is concave up (happy face).

Question 16

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

Answer 16

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

Practically, you should do the following:

1. Find the derivative.
2. Find when f’(x)=0.
3. Choose points to the left and right of each of the answers to part 2.
4. Plug those points into f’(x) to see if the derivative is negative or positive. Then you’ll know that the interval containing that point is decreasing or increasing, respectively.

Question 17

What is a “stationary point”?

Answer 17

Any point that satisfies the equation f’(x)=0. This is also typically a local min/max.

Question 18

What is an inflexion point?

Answer 18

This is the point where concavity changes, which means f”(x)=0.

Question 19

How can you use the second derivative to verify that a point, c, is a maximum or minimum? (Note that you got c by solving f’(x)=0.)

Answer 19