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Calculate the derivative.$$f(x)=x^{5 / 4}+4 x^{-3 / 2}+11 x$$

$\frac{5}{4} x^{1 / 4}-\frac{6}{x^{5 / 2}}+11$

Calculus 1 / AB

Chapter 3

DIFFERENTIATION

Section 2

The Derivative as a Function

Derivatives

Differentiation

Campbell University

Harvey Mudd College

University of Michigan - Ann Arbor

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hi. Today we will be learning how to take that around of off the following the creation Halifax equals X to the power of five over four plus four times X to the power of negative three over two plus eleven x. So first, let's do a small review on how to take the Druid. Let's say that I want to take the derivative off a function, and DD X is just signifies that you want to take the derivative of something. Let's take the derivative of X to the power of any number I the driven of that would just be I Times X to the power off I'm minus what So say that I want to take the derivative off x squared four. Yeah, that's right like that. It would simply be two times X to the power of two minus one, which is just one. No, it's opposing. You had a constant in front of the ex, so let's say we want to take the did it. The directive of C Times X to the power off any number I see being a constant number, like two or three now because see doesn't change with X. We can actually take constant outside of the derivative Vince. They see times the derivative or D T X of X to the power off I. So let's say that we want to take the derivative of two times X to the third power, the derivative of that or DD X Off. This function would be two times three off X two, three minus one or X squared, and that's equal to six Pecs Square. So now, knowing this, we can solve for the derivative off our function above so we can take this in parts. First, let's offer the derivative of X to the power five or four. This would be a prime off X just signifies the derivative FX of Prime Facts equals five over four times X to the five over four minus one five over four minus one is just one forthe soul right like this. Now let's take the derivative of four times X to the power of negative three over, too. Now First will multiply four times negative three over two and will get six Now. Keep in mind that because negative three over to his neck, never going to write minus six because for turn negative. Three over to his negative six times X to negative three over two minus one, which is negative. Five. Over to now, let's write the derivative off eleven X or eleven times. Act to the power of one. Eleven times one is just eleven x to the power ofthe one minus zero is next to the power. Sorry, excellent. Half one minus one is X to the power of zero. And actually carve zero is just one. So it's eleven times one or just eleven. And there you have it. That's your derivative for FX. Thank you for watching.

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