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Calculus - Differentiation - First Principles.
Test Yourself 1.


 

The basic concept of differentiation is that a secant becomes a tangent in its limiting position is outlined in this topic. The law of infinitisimals is a fundamental concept in much of mathematics especially in Calculus.

An understanding of the principles underlying differentiation is important to see where the techniques applied commonly are based.

 

Differentiate each of the following functions by first principles:

1. 2.
3. 4.

 

Remember that your final statement MUST start with . The importance of this statement is that you are stating (to the Marker) that you are defining as being the limit of the expression as h approaches 0. Then you substitute h = 0. If you do NOT do this, you are not defining your derivative.

 

For additional examples to practise with first principles, set yourself questions because you can use the short-cut method to see if you got it correct. Use Questions 1 or 2 and then both 3 and 4 as templates for your questions.

 

Also answer the following related questions:

1. For the function f(x) = 3x2 - 2x + 1:

(i) Find the gradient of the chord joining the points whose x coordinates are 1 and (1 + h) respectively.

(ii) Hence determine the gradient of the tangent at x = 1.

 2.