CALCULUS ABDIFFERENTIATIONBY DANIELLA KRAKUE

The Basics BehindDifferentiation

What exactly is a derivative?The concept of Derivative is one of the most important things you will learn in Calculus.When you solve for a derivative it is called Differentiation.  The derivative is a way to represent rate of change, that is - the amount by which a function is changing at one given point. The derivative is often written using "dy over dx" (meaning the difference in y divided by the difference in x).

Core Topics of Differentiation

1. Derivative as a Function2. Product and Quotient Rul

es3. Rates of Change4. Higher Derivatives5. The Chain Rule6. Trigonometric Functions7. Implicit Differentiation8. Exponential and Logarith

mic Functions

Derivative as a FunctionIn order to solve for the Derivative as a Function we use the Difference quotients. When given a function like the one below , you plug in the given values into the corresponding formulas and solve as approaching the limit.

Find the derivative of the function f(x) = x2

Method 1 Method 2

http://www.mathopenref.com/calcderivfunc.html

More Practice!

Product and Quotient Rule

The Product Rule is expressed in this formula, when given two functions multiplied by one another you take the derivative of the first function and multiply it by the second then you add the product of that to the first function times the derivative of the second function.

The Quotient Rule is expressed in the formula given. When two functions are being divided by one another you take the derivative of the numerator times the denominator, minus the derivative of the denominator times the numerator divided by the denominator squaredhttp://www.intmath.com/differentiation/6-derivatives-pr

oducts-quotients.phpMore Practice!

Rate of Change

http://www.intmath.com/differentiation/4-derivative-instantaneous-rate-change.php

The concept of Rate of Change is an application of the principles of derivatives learned so far. ROC can best be previewed by using this applet on the rate of a melting snowball.http://www.mathopenref.com/calcsnowballproblem.html

More Practice:

Higher Derivatives

The foundation for knowing how to compute higher derivatives is not solely knowing how to use the product/ quotient rules but also the power rule expressed in the equation below.

Examples of Power Rule

http://www.intmath.com/differentiation/9-higher-derivatives.php

More Practice:

The Chain Rule

The Chain Rule is a rule that you use when you can there is a function within a function and you can take the derivative of the outer function.

As shown in the example to the right the chain rule is used with the function x2 where x = (2x + 1). The derivative of x2 would be 2x and the derivative of (2x + 1) =2 (x+1) and when you combine the two you get 4(x+1).

Trigonometric Functions

Now that the most difficult concepts concerning derivatives has been covered on, welcome to the derivatives of trigonometric functions where it simply involves memorizing the corresponding derivative to each trigonometric function.

Practice Lessonhttp://www.math.brown.edu/UTRA/trigderivs.html#top

Implicit Differentiation Implicit differentiation is nothing

more than a special case of the well-known chain rule for derivatives. The majority of differentiation problems we have discussed involve functions y written EXPLICITLY as functions of x. An example of the concept defying the rules would be …

PROBLEM 1 : Assume that y is a function of x . Find y' = dy/dx for x3 + y3 = 4 .

Test you understandinghttp://www.intmath.com/differentiation/8-derivative-implicit-function.php

Exponential and Logarithmic Functions

Finally Exponential and Logarithmic Functions . To master this concept you simply have to remember the following rules and apply them to the given problems.

Derivatives ofExponential and Logarithmic Functions

Test you understandinghttp://people.hofstra.edu/stefan_waner/realworld/tutorials/unit3_3.html