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(a) For what values of $ x $ is the function $ f(x) = \mid x^2 - 9 \mid $ differentiable? Find a formula for $ f'. $(b) Sketch the graph of $ f $ and $ f'. $

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00:40

Frank Lin

Calculus 1 / AB

Chapter 3

Differentiation Rules

Section 1

Derivatives of Polynomials and Exponential Functions

Derivatives

Differentiation

Missouri State University

Oregon State University

Harvey Mudd College

Boston College

Lectures

04:40

In mathematics, a derivative is a measure of how a function changes as its input changes. Loosely speaking, a derivative can be thought of as how much one quantity is changing in response to changes in some other quantity; for example, the derivative of the position of a moving object with respect to time is the object's velocity. The concept of a derivative developed as a way to measure the steepness of a curve; the concept was ultimately generalized and now "derivative" is often used to refer to the relationship between two variables, independent and dependent, and to various related notions, such as the differential.

44:57

In mathematics, a differentiation rule is a rule for computing the derivative of a function in one variable. Many differentiation rules can be expressed as a product rule.

03:40

(a) For what values of $x$…

01:18

(a) Sketch the graph of th…

07:25

03:17

Sketch the graph of $f(x)=…

you know, it's clear a seven inning right here. So we have X squared minus nine. The absolute value we're first going to calculate. We're making a equal to zero. Let me get access equal to positive or negative three. So f of X is not different. Shovel there. We know that Y is equal to X square minus nine. This is gonna be an upward open opening parabola aboard opening crapola and have Ciro's a positive and negative three. So when access less than or equal to negative three and when access bigger or equal to three you have f of X is equal to X square minus nine. We don't need the absolute value signs, since it's already gonna be positive. And when access between negative three and three we get half of X is equal to negative X Square minus nine, which gives us negative X Square plus nine. So when we differentiate for the first part, we get the derivative where X Square minus nine is equal to two x and for access between negative three and three. This is negative. X Square plus nine. We get negative to USC's. So we our answer is it is not different. Chewable at X is equal to positive or negative free and are derivative. Is equal to two X If X is less than negative. Three or ex this bigger than three. Negative to X for access between negative three and positive three. Next, we're going to grab this. So if we grab our original well, look like this then are derivative says that six negative six.

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