Differentiation and Integration

What is the product rule and when is it used?

(The derivative of the first)(The second) + (The first)(The derivative of the second)

REMEMBER IT’S PLUS

Derive the formula for the graph : 6x⁷

42x⁶
is the formula for the slope of the tangent line to 6x⁷

In English, what are f(x) / h(x) / g(x)?

Functions named f/h/g that takes x as an input, does something to it and gives y as the output.

The y you get as the output is the y-value that goes with the x-value you put in to get a point on a graph - (x,y)

For example - if you did this for every number for x from 1 - 100 you would be able to draw the graph of f/h/g between 1 and 100 because you would then have all the points between 1 and 100.

When you derive the formula for a graph what do you get?

A new formula that gives us the slope of the tangent line to that graph, for any given value of x

What is f’(x) (in plain English)

The derivative of f with respect to x
(the name of the formula that gives us the slope of the tangent line for f(x))

What is the chain rule?

(The derivative of the outside without touching the inside) x (The derivative of the inside)

(2x + 3)³ → ( )³ is the outside and 2x+3 is the inside

When should the chain rule be used (some examples)

• ln(2x + 1)

• e^{2x + 1}

• (2x + 1)³

• sin(2x + 1)

• cos(2x + 1)

• √2x + 1
  
  in each, 2x + 1 is the inside and the rest is the outside when applying the chain rule

When is the product rule used?

It is used when two functions are multiplied e.g. (…)(...)

• (3x +2)(2x + 1)

• x² (2x + 1)

• xe^x

• xlnx

• xcosx

• xsinx
  
  where the bold is the first and the non bold is the second

What is the quotient rule?

(The bottom)(The derivative of the top) - (The bottom)(The derivative of the top)
/
(The bottom squared)

REMEMBER IT’S MINUS, AND THE ORDER ITS SUBBED IN

When is the quotient rule used?

(….) / (….)

What is the equation of a line?

y - y₁ = m(x - x₁)

where x₁ and y₁ are a point on the line and m is the slope of the line

What two things do you need to find the equation of a line?

A point on the line and the slope of the line

How do you find the equation of the tangent line to the formula/function f(x) where x = 1

1. Use the f(x) formula to get your point. The point will be (1,y) where the y is the output of f(x)

2. Use the f’(x) formula to find the slope of the tangent line at x = 1

3. Use the point and the slope in the equation of the line formula
   
   

What are stationary/critical points?

They are points on the graph where the slope of the tangent line to the graph = 0

• This means, find the x values you can sub into f’(x) to get 0.

• When the slope of the tangent line is 0 it means the curve is not going up or down, meaning it is the exact point of the peak or dip of the curve
  

How do you know the graph is sloping upwards?

The slope of the tangent line (the derivative / f’(x)) is a positive number

How do you know the graph is sloping downwards?

The slope of the tangent line (the derivative / f’(x)) is a negative number

How do you know the graph is not curving upwards or downwards?

Its derivative equals 0 (these are critical/stationary points)

What nature/description can stationary points have

They can be local maximums, local minimums or inconclusive

How do we determine if a stationary point is a local maximum/minimum using second derivative?

1. Find the x values that makes f’(x) = 0

2. Sub each of the values into f’’(x) (the double derivative)

   1. If the result is negative its a maximum

   2. If the result is positive its a minimum

   3. If the result is 0 its considered inconclusive - try the other method
      
      
      

How do we determine if a stationary point is a local maximum/minimum using trial and error?

1. Find the x values that makes f’(x) = 0

2. Get the value just below and above each of those values (e.g if the value is 4 take 3 and 5 )

3. Sub each of the surrounding values into f’(x) (NOT THE DOUBLE DERIVATIVE)

   1. If the result before is negative and after is positive, its a minimum

   2. If the result before is positive and after is negative, its a maximum

If a third element is being introduced to dy/dx, what formula is used?

dy/dt = (dy/dx)(dx/dt)

What is another way to write f’(x)?

dy/dx

use this when the formula is called y and not f(x)

How do you derive ln(x) - just learn it off

1/x

How do you derive ln(2x + 1) - what rule is it?

This is now chain rule

(1/2x+1)(2)

which is
2/2x+1

What is the derivatieve of e^x

e^x

What is the derivative of e^2x+1

The chain rule

(e^2x+1)(2)
which is

2e^2x+1