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# Formal Derivatives

## by Gideon Weinstein, Online Professor, Mathematics Tutor, Mathematics Education Presenter at Western Governors University on May 28, 2010

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Provides an orientation to the formal limit definition of derivatives along with some examples and extensions.

Provides an orientation to the formal limit definition of derivatives along with some examples and extensions.

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## Formal DerivativesPresentation Transcript

• Formal Derivatives A Short Orientation with Examples By Gideon L. Weinstein Course Mentor Western Governors University Last updated May 2010
• Definition of the Derivative
• The formal definition of the derivative f’(x) of a function f(x) is
• Key Point
• Please note that in this definition, x is a constant and it is h that we treat as the variable as it approaches zero
• The key to these computations is to do as much algebra as possible BEFORE taking the limit.
• Example 1
• Let f(x) = 7x + 3 and formally compute f’(x).
• Please note that in a formal proof, there would be a justification for each the steps taken. That is left as an exercise for the reader.
• Example 1, continued So when f(x) = 7x + 3 then f’(x) = 7.
• Example 2
• Let f(x) = x 2 + 7x + 3 and formally compute f’(x).
• Example 2, continued So when f(x) = x 2 + 7x + 3 then f’(x) = 2x + 7.
• Extensions
• How would the computation change if the first example was 3x + 7 instead?
• How would the computation change if the second example was 5x 2 + 7x + 3 instead?
• What happens with higher-degree polynomials?
• What happens with functions such as sines, cosines, logarithms, and exponentials?