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Find f'(0) and f''(0), if they exist, for the following f ( x ) = \left\{ \begin{array} { c } { x...

Question:

Find {eq}f'(0) {/eq} and {eq}f''(0), {/eq} if they exist, for the following

{eq}f ( x ) = \left\{ \begin{array} { c } { x ^ { 2 } , x < 0 } \\ { x \operatorname { sin } x , x \geq 0 } \end{array} \right. {/eq}

Differentiability at a Point:

When checking the differentiability of a function at a point in its domain, we evaluate the one-sided derivatives of the function at the point. The left- and the right-side derivatives of the function must exist, be finite, and equal for the first derivative of the function to exist at the point. If it exists, the first derivative of the function at the point exists both the one-sided derivatives at the point.

Answer and Explanation:

For all {eq}x \lt 0,\, f(x)=x^{ 2 } {/eq} . So for all {eq}x \lt 0 ,\, f(x) {/eq} is differentiable and using {eq}\displaystyle...

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Derivatives: The Formal Definition

from Math 104: Calculus

Chapter 7 / Lesson 5
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