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Suppose that, in the development of the definition of the derivative, we wrote $(x_{2}, f(x_{2}) \text { for } Q \text { instead of }(x+h, f(x+h)) .$ Show that the definition of. the derivative will then have the following alternate form: $f^{\prime}(x)=\lim _{x_{2} \rightarrow x} \frac{f\left(x_{2}\right)-f(x)}{x_{2}-x}$

Calculus 1 / AB

Chapter 2

An Introduction to Calculus

Section 1

Slope of a Curve

Derivatives

Oregon State University

Harvey Mudd College

Baylor University

University of Michigan - Ann Arbor

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.

30:01

In mathematics, the derivative of a function of a real variable measures the sensitivity to change of the function value (the rate of change of the value of the function). If the derivative of a function at a chosen input value equals a constant value, the function is said to be a constant function. In this case the derivative itself is the constant of the function, and is called the constant of integration.

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Computing the derivative o…

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02:03

Use the definition$$

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Use the limit definition …

01:21

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03:37

Use $f^{\prime}(x)=\lim _{…

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01:38

This is the definition of the derivative. And now when that X two equals X class age, this is equivalent to H Echoes X two Miners X. And the edge goes to zero X two girls 2 x. So we just substitute the definition of the derivative. Yes, X two girls 2 x and X plus H echoes X two and eight equals X two minus X. This is just the result we want.

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