Category Archives: MAT 131

L’Hospitals Rule – Calculating Limits using Derivatives.

Use this for solving problem cases, where the limit on the right side exists or is +-infinity. These are problem Cases.

This is L’Hospital rule, used for the problem cases above.

This is how to rearrange your limits so you can use L’Hospital’s rule and still find the limit even though it has an indeterminate form. The situations are: Indeterminate quotient, Indeterminate product, Indeterminate difference and indeterminate powers.

Also, a note on derivatives and the chain rule. This is what you do with constants. Including e, which is a constant.

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Review for Midterm #2

f is continuous if

The limit as x approaches a exists for f(x), the limit as x approaches a equals f(a) and it is continuous along the interval if true for every point a in the Interval.

f is differentiable if

The limit as h as f evaluated at x plus h minus f evaluated at x over h exists. Along an interval if limit exists for every point a in the interval.

A differentiable function is always continuous, but not the other way around. A continuous function can not have a derivative at a point (Break or sharp corner)

GRAPHING FUNCTIONS!!!!!!!!!!

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Logarithmic Differentiation

When trying to solve a problem like y = (ln x)cos x use Logarithmic differentiation so you can use the product rule.

And a Trig identity I just recently grasped the usefulness of.

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Implicit Differentiation, Tangent Lines, Inverse Trig Functions

Find the tangent line at point (3,2)

Now that we know dy/dx, solve for the slope using the given values for (x and y)

Now use the Tangent Line formula

Next Problem

Now to find the tangent line for this problem, with the coordinates (2,0)

A brief aside

Prove derivative of ln(x) using Implicit Differentiation

  1. The Problem
  2. Write in full form, including y
  3. Use log properties to arrange into x = form
  4. Take the derivative with respect to the change in x from both sides
  5. The derivative of “e to the y” remains “e to the y”, the derivative of x is one.
  6. Rearrange to solve for the derivative dy/dx.
  7. Substitute for y.
  8. “e to the ln of anything” is equal to the anything, in this case anything is x.

Find the derivative of arcsin x.

  1. The Problem
  2. Rearrange the problem using Trig
  3. Take the derivative from both sides
  4. x is equal to 1
  5. Solve for dy/dx
  6. Substitute for y in the solution.
  7. The professor kept going and somehow (using Trig?) managed to bring the answer to this.

This is what he did:

Not quite sure how cos y equaled x

Increasing and decreasing functions

For sin to be invertable, you need to restrict the domain to -pi/2 to pi/2
To see how a function looks, determine the invervals above and below zero. To do this solve for x.

You can look at the derivaties to decide in a function is concave up or concave down.

Second derivative is negative, concave down. Second derivative is positive, concave up

READ problem 2.7 #43 from the textbook.

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Extra Derivative Tips

General Concepts which are useful

Some extra Derivative Tricks

Derivatives of Trig Functions

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Trigonometric Function Graphs

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Derivatives and Implicit Differentiation

How do you find two tangent lines passing through a point? The point is not a point on the line! So decide on an arbitrary point on the line and call it f(a). Then solve for the slope m in terms of a (Find f’(a)). Then plug in your values and solve for the tangent line at the two points.

Derivative Example Problems

Rule: Apply log rules when you see logs!!!!!! (You want to expand as much as possible)

Implicit Differentiation

We briefly started to talk about implicit differentiation. So a function is y=f(x). or “y as described by function f with respect to point x equals”. And this seems to be the key here. So we started using this new notation for derivatives, instead of the usual f prime. To start, look at a very easy example, x2.

y equals x2 which is the same as, f(x) equals x2, which says that y = f(x).”

“The derivative of the function which equals y with respect to point x equals f prime, which is equivalent to the derivative of x2 equal 2x.”

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Rules on Derivatives – Part 2

Product Rule

Start by figuring out which is f(x) and which is g(x). Then find the derivatives for f(x) and g(x). Then write out the product rule in order f,g,h,j,k… Etc however many functions are in the product. Then make one of the functions in each multiplication a derivative.

Some examples to practice:

Quotient Rule

So the quotient rule is a little funny. It has the g function squared on the bottom. I wonder what would happen if there are two functions on the bottom?

Quotient Rule Examples:

Combining the Product Rule and the Quotient Rule

First, do the Quotient Rule, then do the product rule.

Chain Rule (Review for accuracy)

The Chain rule is used for composite functions. IE: One function (inner) is passed to another function as an argument.

myFunction(anotherFunction());

So for the Chain rule, this would be

myFunctionDer(anotherFunction()) * anotherFunctionDer()

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Derivative – Rules Part 1

I found out the answer to why there are two very similar definitions of the derivative. One has x-a on the bottom, with a limit x → a. The other one has h on the bottom, with a limit h → 0. So these two things are the same, just different ways of thinking about the same concept. h represents the difference between x and a, so the smaller the difference gets the closer h is to 0. The official definition is the one with h.

A function F is differentiable on an interval if F is differentiable for every point a.

Key Properties of Differentiability
1. F is differentiable if F is continuous
2. F is differentiable if F is smooth.

For |x| write it in bracket form!

Q1: Is f(x) continuous? Yes because a limit exists for all point a.
Q2: Is f(x) differentiable? Use the definition of derivative (h → 0) and see if the same from left and right. This derivative from the left and the right is not the same, it is -1 and 1. F is not differentiable at 0.

Rule
Tan, always break apart.

Property – Write this on a card
1. If f’(x) < 0 is decreasing near x and inverse: a. f’’(x) > 0 f is concave up.
b. f’’(x) < 0 f is concave down.

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The Beginnings of Derivatives



The meaning of f’(a) [read “F Prime”] is the slope (tangent) at point a.

There are three main ways of thinking of the derivative.
It is the slope of a line
It is the Instantaneous velocity
It is the rate of change

Some Basic Derivative problems

(Practice and be able to explain these, using the Derivative formula (Limit h approaching 0…)

Find the Tangent line at point x = 2

y minus initial y equals the slope times x minus initial x.

Domain in f(x) and f’(x)
The Domains of a function and it’s derivative are not always equal to each other, the Domain of f(x) is greater than or equal to the domain of f’(x). We say that f is differentiable in f if f’(x) exists. To prove f’(x) exists, the following three must exist.


We say that f is differentiable if f’(a) exists for all a in the domain.

The smooth graph is differentiable. A graph with corners is not. Eq 4, above, is an example of a graph which is not differentiable at 0. To be differentiable implies continuity.

PROVING CONTINUITY
A function is continuous if the following two limits exist, and when multiplied by each other equal something.


Meaning of Derivative
The Derivative is the line of the slope. When it crosses an axis you have reached a minimum or maximum value. The line of negative slope is going down. The line of positive slope is increasing. Look at Tangent lines, they approximate the graph and tell the direction up or down.

In Conclusion
f’(x) is negative, x is decreasing near x.
f’(x) is positive, x is increasing near x.

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