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In calculus, the difference quotient $\frac{f(x+h)-f(x)}{h}$ of a function$f$ is used to find a new function $f^{\prime},$ called the derivative off. To find$f^{\prime},$ we let $h$ approach $0, h \rightarrow 0,$ in the difference quotient. For example, if $f(x)=x^{2}, \frac{f(x+h)-f(x)}{h}=2 x+h,$ and allowing$h=0,$ we have $f^{\prime}(x)=2 x$$$\text { Given } f(x)=6 x+\sqrt{x}, \text { find } f^{\prime}(x)$$

$6+\frac{1}{2 \sqrt{x}}$

Precalculus

Algebra

Chapter 1

Functions and Their Graphs

Section 1

Functions

Algebra Topics That are Reviewed at the Start of the Semester

Oregon State University

Baylor University

University of Michigan - Ann Arbor

Lectures

01:43

In mathematics, a function is a relation between a set of inputs and a set of permissible outputs with the property that each input is related to exactly one output. An example is the function that relates each real number x to its square x^2. The output of a function f corresponding to an input x is denoted by f(x).

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15:15

In calculus, the differenc…

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04:36

Find the difference quotie…

02:14

Given f(x) =x? X + 1 simpl…

in this problem we are given the function if or fixed equal to six express square topics and we are asked to find a prime affects which is actually the little bit of heaven affects. You can find this using the different caution formula And this is basically a 4th explosive charge minus four fix over hitch. So in this difference caution If you let hedge approach, zero then this becomes a formula for a prime office. So this is equal to F prime of X. When Hedge approaches zero. So let's see how to find out this F. Prime of X. So for this we have to first calculate the F. Of X. Presage minus of or fix over hedge. So let's find out this difference caution. So we have to find a full of explosives. We have to replace X by explicit in this expression all access to be replaced by explicit. So when we do that we will get six times of X plus hedge right plus explicit under the rope. So we have done this f of explicit minus For fixed. So we put therefore affects our cities. Since it is negative, these two will become negative. So therefore this will become -66 minus of square root of X. Everything under the hitch. Let's distribute to the 6th and we can simplify this. Let's do that. This is 66 plus six H. Plus this will remain as it is quite a lot of X plus edge minus 66 minus square root of X. All over. Which let's see if we can cancel some of the terms we have passed 266 and 96 of these two will get cancelled. So that's the only thing which we can cancel now and then we are left with the six H plus squared off explosives minus square topics. So we can write on this as six H plus square root of X. Plus hedge minus square root of eggs all over hedge. When I do that we can split this into two fractions as six. Such over hitch. And this term another term over hedge so that we can simplify a little bit further. What is that? That this hedge and this hedge will get cancelled and we are left out with six. Lift out with six plus express hedge minus square root of eggs all over hedge. To simplify this radical expression, we have to multiply both the numerator and denominator by the conjugate. After this expression expressive minus squared off X squared off X. The conjugated of this is the similar expression with a plus sign. We have to we have to put the plus sign over here. So let's do this in the next step. So this is 6126. We have the 64 here. This will become this will remain as it is quite a bit of explicit minus square root of X. All over hedge, multiplied by its conjugate conjugate of this expression. So it is sequined too explicit. This minus we have to put plus and squadra topics and we have to do it with the same quantity squared off, express edge plus square or topics. So the next time we are going to multiply this quantity at least that if you can consider that this term as mm And this term as B. We have a -7. And here also we have the same term a. And here also we have the same term be. So therefore this is basically a product of a minus B times A. Plus. We and when we do that we can at least the reference of caution. Uh we can at least Difference of Squires identity. That is a -7 times a plus weeks. This is actually equal to a square minus B squid. So it's a place. This difference of squares form long. We're here. So therefore the six will remain as it is. So we have to Squire this Squire it of explosives when you do that, the Squire it will go off. So therefore this will become explicit. And since we have negative we put the negative over here and be square square square off excess. Be so in the spirit of the square it will go off. So therefore this will become and tracks all over H. Times the denominator. Well we have this factor so this we have to multiply with this factor will not have to change anything. So this will become scared of explicit plus square root of X. What is that? We can cancel this X. And a negative X. And when you do this, this too will get cancelled. And we will also help to cancel this act this H. And this age will get cancelled. So let's write down. So finally we have six plus. So the numerator, we have one all over the denominator. We have explosives plus score a graphics. So that's the only time that we have in the denominator. And finally, as part of the definition, we have to let hedge approach zero. So therefore this is perfectly we're finding from F prime of X. We have to let hatch approach zero. So let's do that here. When you let hatch approach zero, This will become zero. So therefore will be getting this will be squared of X plus squared off exports are like them. So we'll get ah square it off two times of school topics so therefore we can write down these six plus one over two times of square tropics. So there's the answer for this question, and we have phone road F prime of X for the function, the four fax equal to six X plus squared off X.

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