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Use your knowledge of the derivative to compute the limit given.$$\lim _{h \rightarrow 0} \frac{(x+h)^{2 / 3}-x^{2 / 3}}{h}$$

$$\frac{2}{3 x^{1 / 3}}$$

Calculus 1 / AB

Chapter 2

An Introduction to Calculus

Section 2

Derivatives Rules 1

Derivatives

Oregon State University

Harvey Mudd College

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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So the underlying theme in this problem is the limit definition of the derivative. Eso I'm just copy this down to the two thirds power minus X to the two thirds power all over H. This is the definition of the derivative. Uh, if you can identify what FX is now, to me, it's really easy because I've will practices a lot because it's f of X plus h minus ffx. So this piece right here must be effects. Eso basically what it boils down to is what's the derivative of that blue thing and just wrote down, Will you bring that two thirds in front? Subtract one from that exponents? Well, two thirds minus three thirds would be negative. 1 30 Now, I would let my students leave their answer like this. Other teachers might say, Oh, you could go ahead and rewrite that as the cube root of X in the denominator, which it is the same thing. Um, it's really a teacher preference at this point, you know, either circled answer is correct.

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