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Assuming each limit exists, compute them. (a) $\lim _{h \rightarrow 0} \frac{f^{\prime}(x+h)-f^{\prime}(x)}{h}$(b) $\lim _{h \rightarrow 0} \frac{f^{\prime}(x+h)-f^{\prime}(x-h)}{h}$

(a) $f^{\prime \prime}(x)$(b) $2 f^{\prime \prime}(x)$

Calculus 1 / AB

Chapter 3

Applications of the Derivative

Section 3

Concavity and the Second Derivative

Derivatives

Missouri State University

University of Michigan - Ann Arbor

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.

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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Find each limit.$$…

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assuming each limit exists, we want to compute them. Um And we see that the first one is going to be um f prime of X because we know that the derivative of F of X is going to be F of X plus H. My type of acts over H where we take the limit as HBO's zero. However, if we end up making this F prime of X minus F prime of X, what we end up seeing is that this right here is going to be um the 2nd derivatives of fxff double convex. And then the other one, we see that based on what given to us, We're going to end up getting two times second derivative of X. So that's going to be our final answer.

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