Use Figure 3.14 to estimate the derivative, or state it does...

Use Figure 3.14 to estimate the derivative, or state it does not exist. The graph of $f(x)$ has a sharp corner at $x=2.$ (Check your book to see figure)
Let $u(x)=g(f(x)) .$ Find:
(a) $u^{\prime}(1)$
(b) $u^{\prime}(2)$
(c) $u^{\prime}(3)$

Key Concepts

Composite Functions

Composite functions are functions formed by applying one function to the results of another. When dealing with a composite function, such as u(x) = g(f(x)), the output of the inner function f(x) becomes the input for the outer function g. This concept is fundamental for understanding how changes in the inner function can affect the overall behavior and derivative of the composite function.

Chain Rule

The chain rule is a formula used to compute the derivative of a composite function. It states that the derivative of u(x) = g(f(x)) is u'(x) = g'(f(x)) * f'(x), provided that the derivatives of both f and g exist. This rule is essential for breaking down the derivative of a complex function into simpler parts that can be individually evaluated.

Differentiability

Differentiability refers to the property of a function having a derivative at a particular point. A function is differentiable at a point if it has a well-defined tangent at that point. If even one of the component functions in a composite function is not differentiable at a given point, then the composite function also fails to be differentiable at that point.

Sharp Corners and Non-differentiability

Sharp corners occur on the graph of a function where the direction of the curve changes abruptly. At these points, even though the function may be continuous, the derivative does not exist because there is no unique tangent. This concept is important as it illustrates that continuity of a function does not always imply differentiability.

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