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The Derivative Function- DM4

Objectives

  • Define the derivative function of a given function.
  • Graph a derivative function from the graph of a given function.
  • State the connection between derivatives and continuity.
  • Describe three conditions for when a function does not have a derivative.
  • Explain the meaning of a higher-order derivative.

Summary

The limit definition of the derivative, f ′ ( x ) = lim h → 0 f ( x + h ) − f ( x ) h , produces a value for each x at which the derivative is defined, and this leads to a new function y = f ′ ( x ) . It is especially important to note that taking the derivative is a process that starts with a given function f and produces a new, related function f ′ .

There is essentially no difference between writing f ′ ( a ) (as we did regularly in Section 1.3) and writing f ′ ( x ) . In either case, the variable is just a placeholder that is used to define the rule for the derivative function.

Given the graph of a function y = f ( x ) , we can sketch an approximate graph of its derivative y = f ′ ( x ) by observing that heights on the derivative's graph correspond to slopes on the original function's graph.

absolute value graph
F i g . 1

In Figure 1, we encountered some functions that had sharp corners on their graphs, such as the shifted absolute value function. At such points, the derivative fails to exist, and we say that f is not differentiable there. For now, it suffices to understand this as a consequence of the jump that must occur in the derivative function at a sharp corner on the graph of the original function.

See the Desmos Demonstration.

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