Blog for October 23

Overview of the week of 10/17-10/21:

Monday: We reviewed derivatives of exponential and logarithmic functions

Tuesday: PSAT day

Wednesday: NO class

Thursday: Chain rule/derivatives of logarithmic and exponential functions assessment

Friday: Started covering our last type of derivative (!!!), implicit differentiation (derivatives of non-functions)

Implicit differentiation- writing the derivative when y isn't alone, or when the equation can not be solved for y. An example of when implicit differentiation must be used is x2+y2=1

All year we have been finding the derivative of functions in respect to x (without knowing it)

When finding the derivative of an equation like y=x+1 we would find y'=1 or dy/dx= 1, where dy/dx is the derivative of y and 1 is the derivative of x+1.

Now that y is not always alone, and can be on both sides of the equal sign dy/dx has a much bigger role in our derivatives.

Now that we are finding derivatives of y terms that are more than just y (ex. y2, 3y, 17y2) dy/dx is always part of our equations, because it's part of the derivative of y.

When using implicit differentiation it is important to remember to always write dy/dx because it is part of the derivative of the y term

example: The derivative of x=4y is 1= 4(dy/dx)

• 1 is the derivative of x
• 4(dy/dx) is the derivative of 4y

Example from class:

The steps we took when solving this were:

1. Take the derivative
2. Collect dy/dx terms on one side of the = (anything else should be moved to the other side)
3. Factor out dy/dx
4. Divide to get dy/dx alone on one side of the =

**Helpful hint: distributing almost always makes these problems easier

Here is another example from class. This problem involves more than just finding the derivative of an equation: 

Implicit differentiation can also be used to find the second derivative of an equation.

1. Find the first derivative normally, using implicit derivation 
2. Find the second derivative (almost always involves the quotient rule)

Here is an example from class:

**Don't forget: when writing the second derivative dy/dx becomes d2y/dx2

Blog for October 16

Overview
This week's material focused on the eighth derivative rule regarding derivatives of exponential and logarithmic functions with introduction to the number e. The material also introduced derivatives of inverse trigonometric functions.

RULE #8 - Derivatives of Exponential and Logarithmic Functions

Derivatives: "e" as an Exponential Function
In the exponential section of derivatives, we were introduced to the numerical value e. The derivative of e in an exponential function will always equal itself, as written below.

• dy/dx (ex) = ex

Applying "e" to Several Derivative Rules

*Product Rule


*Quotient Rule


*Chain Rule


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Derivatives: Exponential Functions with Values Other than "e"
The derivative of an exponential function that is not ex (i.e. ax) is:

• dy/dx (ax) = ln (a) * ax

*Proving the Derivative Formula


"a" is only representative of a value. Numerical values can be substituted for this variable, and the same steps can be followed to find the derivative. An example of this process with numerical values is listed below.

*Derivative of an Exponential Function Other Than "e"

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Derivatives: Natural Log Functions
The derivative for natural log functions is given and proven as listed below:

• dy/dx [ln(x)] = 1/x

*Proving the Derivative Formula

Now we can apply this formula to natural logs of numerical values opposed to only working with variables.

*Derivative of a Natural Log Function

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Derivatives: Logarithmic Functions

Logarithmic functions have a derivative formula that follows the format listed below:

• dy/dx [loga(x)] = 1/ln (a) * x

*Derivative of a Logarithmic Function

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Derivatives of Inverse Trig Functions

Aside from working with exponential and logarithmic functions, this week's material encompassed an introduction to derivatives of inverse trig functions for sine, cosine, and tangent. Each of these trig functions has a specific derivative formula.

Derivatives: Inverse Sine

The derivative formula for finding the derivative of inverse sine is provided below with an example highlighting how the formula can be applied. 

• dy/dx [sin-1(x)] = 1/√(1-x2)

*Derivative of Inverse Sine Function

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Derivatives: Inverse Cosine

The formula for the derivative of inverse cosine varies slightly in comparison to the formula for the derivative of inverse sine. One important thing to note is that inverse cosine is negative and inverse sine is positive. Remember this to avoid using the wrong formula. 

• dy/dx [cos-1(x)] = -1/√(1-x2)

*Derivative of Inverse Cosine Function

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Derivatives: Inverse Tangent
Finally, the derivative formula for the inverse of tangent follows:

• d/dx [tan-1(x)] = 1/1+x2

*Derivative of Inverse Tangent

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FINAL TIPS

Don't forget several log properties that are important to use:

• ln(ab) = ln(a) + ln(b)
• ln(a/b) = ln(a) - ln (b)
• ln(ax) = x * ln(a)

Blog for October 10

Overview-
This week in class we focused primarily on the Chain Rule. It is the 7th rule of derivatives and is one of the most important.

Equation-

Application-
Every time there is a function nested inside of another function the chain rule is extremely useful. It is used to link parts of equations together and for differentiating complicated equations. It allows people to take derivatives of much more complex problems.

Explanation-
To take the derivative you have to look at the most outside function, usually an exponent. Just pretend that the inside is just an x as usual and take the derivative and leave the inside portion unchanged, but still copy it down. Then multiply that answer by the derivative of the stuff on the inside that you previously ignored. That is the basics of the rule, but this process may have to be repeated depending on the extent of the problem.

Practice Problems- 

Using the Rule Twice in Same Problem-

  








Finding dr/d(theta)

More Complicated Problem-

Using Values From a Table-

Tips-

- Always work from the outside in

- Anytime there is a nested function, think to use the chain rule

Blog for October 2nd

This week in Calculus we continued working on derivatives. We learned about position, velocity, speed, acceleration, and higher order derivatives. We also learned the four remaining derivative rules.


Position, Velocity, Speed, Acceleration:




Important Points:

• Velocity is first derivative (slope of position), acceleration is second derivative (slope of velocity)
• To find average velocity, find slope of secant line
• To find instantaneous velocity, find derivative

Derivative Rules:
1.

This can be proved by:


2.

This can be proved by:



3.

This can be proved by:


4.
 
This can be proved by:



Practice Problems Using these Rules:
1.


2.