The Derivative: “Introduction to Techniques of Differentiation” Section 2.3 The Derivative: “Introduction to Techniques of Differentiation”

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Summary In this section we will develop some important theorems that will allow us to calculate the derivatives more efficiently. We will only use the “long way” (definition of the derivative) to develop the new shortcuts. 

The Derivative of a Constant If you consider the graph of y=2 (for example), or y = any constant, it is a horizontal line whose slope is zero (see graph page 155 if you have a question). Since the derivative of y=2 equals its slope, the derivative must be 0. y’ = f’(x) = 0

Derivatives of Power Functions We have taken the derivative of linear (x1), quadratic (x2), and cubic (x3) functions in previous sections and talked about the “shortcut rule” or power rule. There is a more formal proof in the middle of page 156.

Derivative of a Constant Times a Function

Derivatives of Sums and Differences We can use a proof very similar to the one on the previous slide (see page 158) to find the derivatives of sums and differences. Which just means that the derivative of a sum/difference equals the sum/difference of the derivatives (as it did with limits).

Example Using These Theorems

Higher Derivatives The second derivative is the derivative of the first derivative and all of the same rules apply. As long as we have differentiability, we can continue the process of differentiating to obtain third, fourth, fifth, and higher derivatives of a given function.

Example of Higher Derivatives

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