Blog for September 25th

This week in Calculus we worked on the Derivative Rules. Below are some detailed notes about all six of the rules that we learned this week that make finding derivatives easier.

1.) The Power Rule

The Power Rule was the most important rule that we learned this week, and this rule explained how to find the derivative of a function instead of using the limit rules. In short, the Power Rule uses the equation for a derivative function to prove that there is an easier and faster way to find the derivative of a function.

Above is an explanation as to how we got the formula f '(x)=nx^n-1

This is an example of the Power Rule.

2.) The Constant Multiple Rule

This rule means that if there is a constant in front of the x, then you will multiply the constant by the exponent.

 This is an example of the Constant Multiple Rule.

3.) The Sum and Difference Rule

We would use this rule if the function was a polynomial for example, or if it had a constant. This rule states that adding multiple functions together would be the same as adding the derivatives of the same functions together, and the same goes for subtraction.

   This is an example of the Sum and Difference Rule.

4.) Derivative of y= sin(x) and y= cos(x)

We use the derivative of sine and cosine graphs to determine what the derivative graphs would look like for some of our trig functions. We found that the derivative graph of sine is cosine and the derivative graph of cosine is negative sine.
   Here are the graphs proving the rule.

    Here are some examples of the rule.

5.) The Product Rule

The Product Rule gives us an equation on how to find the derivative when there are two functions multiplied by each other. The equation would be the first function multiplied by the derivative of the second function plus the second function multiplied by the derivative of the first.


    Here is an example of the product rule.

6.) The Quotient Rule

The Quotient Rule gives us an equation for finding the derivative of two functions divided by each other.

    Here is an example of the Quotient Rule.

* One great tool to learn to help you memorize the Quotient Rule is:

low, D high; high, D low/low squared

D represents the derivative of and low means the bottom function and high represents the top function.

Weekly Wrap Up, 9/18

These are the major calculus concepts covered during our first week of class.

1. The derivative of a function refers to the slope of a line drawn tangent to a function at a particular point.  In order to find the value of the derivative of a function at a particular x-value, you can use the following limit definition:

This definition starts by finding the slope of a secant line between x = a, the point where you want to find the derivative, and some other point on the function.  Then, by taking the limit as x approaches a, we are able to consider what happens when the two points become infinitely close together, turning the secant into a tangent.

2. It is also possible to find a formula for the derivative of a function at any point.  This is know as the derivative function.  The limit definition of the derivative function is:

This is the major definition for the derivative you should know.

3. You can find the equation of a tangent line to a function at a certain x-value if you know the y-value and the derivative of a function as a certain x-value.

For example, if f(2) = 5 and f'(2) = -3, then the equation of the tangent line (in point slope form) is

y - 5 = -3(x - 2)

4. A normal is a line perpendicular to a tangent at a particular point.  The process for writing the equation of a normal is the same as writing the equation for a tangent, except that once you find the slope of the tangent, you need to take the negative reciprocal of the tangent slope to find the slope of the normal.  So, if f'(2) = -3, then the slope of the normal is 1/3.

5. The slopes of an original function f(x) translate into y-values on the graph of a derivative f'(x). 

This means:

• When the graph is increasing, the derivative graph is above the x-axis
• When the graph is decreasing, the derivative graph is below the x-axis
• When the graph has a min, max, or some other kind of horizontal tangent, the derivative graph will have a zero/x-intercept.