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This article explores the differentiation of algebraic combinations of functions, focusing on constant multiples, sums, and differences. We will discuss the ease of computing derivatives using the definition, and tackle more complex functions that challenge this method. The key question is whether we can derive the derivatives of simple functions and use that knowledge to find the derivatives of their algebraic combinations without reverting to the difference quotient. We will demonstrate how to differentiate these combined functions effectively.
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Differentiating “Combined” Functions ---Part I Constant Multiples, Sums and Differences
Algebraic Combinations • We have seen that it is fairly easy to compute the derivative of a “simple” function using the definition of the derivative. • More complicated functions can be difficult or impossible to differentiate using this method. • So we ask . . . If we know the derivatives of two fairly simple functions, can we deduce the derivative of some algebraic combination (e.g. the sum or difference) of these functions without going back to the difference quotient? Yes!
The Derivative of a constant times a function Can we do this?
The Derivative of the Sum of Two Functions Can we do this?
