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Use your knowledge of the derivative to compute the limit given.$$\lim _{h \rightarrow 0} \frac{\sqrt[3]{8+h}-2}{h}$$

$$1 / 12$$

Calculus 1 / AB

Chapter 2

An Introduction to Calculus

Section 2

Derivatives Rules 1

Derivatives

Campbell 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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alright. Using what we know about limits and derivatives here, we're gonna be computing the limit given We can see from this that because of the form it's in, we're able to draw out the fact that X is equal to eight. And that end is equal to one third and being equal to one third. Because by taking the cube root here of eight plus h, that's as if we were writing it an exponent form. We could do eight plus h to the one third, making it very clear from this form written in blue, that one third is our end value and eight is our X value. So because of that, we're able to use the form end times X and minus one, and that will give us the limit. So just plugging in the values that we just found for N and X we get eight times one third we get one third times eight to the one third minus one. If you plug that into a calculator, do it by hand. You will find that our limit is 1/12

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