Derivatives

The formula to get the derivative of a function using first principle?

What is the definition of a differentiable function?

Is it true that is a function is differentiable at a, then the function is also continuous at that point?

Yes

Is it true that is a function is continuous at a, then the function is also differentiable at that point?

No

Proof if a function is differentiable at point a, it is also continuous at that point

What are the 3 cases in which a function is not differentiable

What is the binomial theorem formula?

What is the derivative of a constant?

What is the combination formula?

What is the permutation formula?

What is the factorial formula?

What is the derivative of power

What is the constant multiple rule (differentiation)

What is the sum and difference rule (differentiation)

What is the product rule (differentiation)

What is the quotient rule (differentiation)

What is the derivative of a sin function?

What is the derivative of a cos function?

What is the derivative of a tan function?

What is the derivative of a cosec function

What is the derivative of a sec function

What is the derivative of a cot function

What is the chain rule (derivative)

What are the condition for chain rule?

Both the outer and inner functions must be differentiable.

What is implicit differentiation

What is the first derivative of a parametric equation

What is the second derivative of a parametric equation

Draw a inverse sin function

Get a inverse function a sin function

What is the derivative of a inverse sin function

Draw the inverse cos function

Draw the derivative of a inverse sin function

What is the derivative of a inverse cos function

Draw the inverse tan function

What is the derivative of a inverse tan function

Draw the graph of the inverse tan function

Draw the graph of an inverse cosec function

y ∈ [-π/2,0) ∪ (0, π/2] and x ∈ (-∞,-1] ∪ [1,∞]

What is the derivative of an inverse cosec function

Draw the derivative of an inverse cosec function

Vertical asymptote at x = -1 or x = 1

What is the derivative of an inverse sec function

Draw the function of an inverse sec function

y ∈ [0/π/2) ∪ (π/2, π]

Draw the inverse cot function

y ∈ (0, π)

What is the derivative of the inverse cot function

What is the derivative if a natural log function

What is the derivative of a natural function with an inner function (chain rule)

What is the derivative of an ln|x|

What is the derivative of log function with a base of b

What is the derivative of a log function that has both an inner function and a function as a base

What is the derivative of an exponential function with a base of b?

How to rewrite a exponential function of a base of b with a base of e?

How to rewrite a exponential function where both the base and the exponent is a function to a exponential function where just the exponent is a function?

Express Euler’s number as a limit

What the formula for linear approximation?

How to know if you are overestimating or underestimating in your linear approximation? (graphically)

What is the formula for linear approximation using differential

What is the difference between dy and ∆y and dx and ∆x in differential

dy represents the differential change of y based on the derivative (change of the linearization) , while ∆y represents the actual change in y over an interval. Similarly, dx is the differential change in x and ∆x is the actual change in x over an interval.

Steps to solve related rates

What is absolute maximum and minimum values?

Does absolute maximum and minimum values include end points?

Yes

What is a local minimum and maximum value?

Does local maximum and minimum values include end points?

No

What is the Extreme value theorem

Condition

• Continuous ∀ x ∈ in the interval

• Interval as to be a closed interval

Why local maximum and minimum be at end points?

You do not know the trend of the graph outside the interval

Possible cases where you can not use Extreme value theorem

What is the Fermat’s theorem

Conditions:

• At point c → the graph f is a local minimum or maximum

• The graph f is differentiable at point c

∴ f’(c) = 0

Cases where you can’t use Fermat theorem

What is a critical value

Does a circle have an absolute/local max/min value?

No

Is it possible for a function to not have a limit at c but point c is a absolute/local max / min ?

What is the closed interval method

What is Rolle’s theorem

What is the mean’s value theorem

What is L’Hôspital rule?

What do you do when you have an indeterminate product?

What do you do when you have an indeterminate difference?

What do you do when you have an indeterminate power?

What does it mean when an expression is indeterminate?

It is an expression that does not have a definite value (showing a competition between two super powers and depending their individual strength → one may win or there will be a tiebreaker)

What is the increasing / decreasing test

What is the first derivative test

What is a second derivative?

What is the concavity test?

What is the point of inflection?

What is the second derivative test

What is the guideline to do graph sketching?

Domain

Find set of values of x for which the function is defined

Intercept

• x-int (setting y = 0)

• y-int (setting x = 0)

Symmetry

(this is not needed)

• f(-x) = f(x) → symmetric by the y-axis

• f(-x) = -f(x) → symmetric by the origin

• f(x + h) = f(x) → symmetric by the period

Asymptotes

• Horizonal

• Vertical

• Slant

Interval of increase / decrease

Use the I/O test

Local max / min

• Find critical value

• FDT

• SDT

Concavity & point of inflection

CT

Sketch the curve

Done