Our Discord hit 10K members! 🎉 Meet students and ask top educators your questions.Join Here! # 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, and the derivative of velocity with respect to time is acceleration. The concept of a derivative has been generalized to other contexts than just functions of a single real variable. For instance, the derivative of a function of several real variables is a function of several real variables, and the derivative of a function from Rn to Rm is a function from Rn to Rm.

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Okay, so we want the X values for which this function, which should look familiar, has a horizontal tangible. So in other words, we want to find X values such that F prime of X is equal to zero. So, in other words, this slope derivative is the slope of the tangent line. The slope of those tangent lines is your But remember that we already found at the prime of X In the previous example, F prime of X was equal to three X squared minus six x minus 24. Okay, we just did that by using the definition of the derivative. So if I want f Prime of X to be equal to zero, I'm really looking to solve three X squared, minus 66 minus 24 equals zero. So one thing we could do right off the bat, it's factor out of three, and that will help us try to see if we can factor. Yes, what is a quadratic function? So that's minus two. X factor out of three and then minus eight goes here. Okay. And I see that there's actually factors, so this is three times X minus four times x plus two. So, of course, one of these three things needs to be zero for the product to be zero using zero product property. And, of course, three can't be 03 is just three. So either excess for or X is negative. So these air the X values which our function has a horizontal dangerous. Georgia Southern University

#### Topics

Differentiation

Applications of the Derivative

Integrals

##### Top Calculus 1 / AB Educators ##### Catherine R.

Missouri State University  ##### Kristen K.

University of Michigan - Ann Arbor ##### Michael J.

Idaho State University