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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$$$\text { Given } f(x)=x^{3}+x, \text { find } f^{\prime}(x)$$

$f^{\prime}(x)=3 x^{2}+1$

Precalculus

Algebra

Chapter 1

Functions and Their Graphs

Section 1

Functions

Algebra Topics That are Reviewed at the Start of the Semester

Missouri State University

Campbell University

Idaho State University

Lectures

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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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In calculus, the differenc…

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Difference Quotient Find $…

in this problem we find a primary effects When a four fixes given us X cube plus X. So we find this using the difference caution for a prime optics that is a frame of access. Written as a form of explicit minus F. Or fix all over the edge. So this is basically the difference caution. The next step to find A for fix plus hedge. All that we do is replace X. S explicit. So therefore this becomes express hedge cube plus X plus hedge -4 Fax. We lay down there for success. It is this will become X cubed plus X within the brackets and then all over hedge. And we have to expand this explicit all cube. You can do this using the algebraic identity. A plus B quantity cube Is equal in two. Thank you. Plus BQ plus three A squared B plus 38 three A B squared. So therefore this explicit all cube will become X cube also become as X cube plus uh HQ Plus three x squared hedge Plus three X. Head Square. So now that we have expanded this explicit all cube. Then we ride on this explosive charge as it is. So we don't ask explicit. And then we expand this bracket minus X q minus X. All quality divided by hedge. Let's clean this up. So we have X cube minus sq can cancel this. We have post two X and NATO X. We can cancel this. All right straight down after we bring this up. So we have the remaining terms in the numerator. As HQ Plus three x squared hatch plus three X. It's clear. And then the hedge tomb hitch. This one is all over by Hitch. As you can see that the numerator all terms has hedged. Which which we can divide by hedge. So therefore this can be taken as it's squared Plus three x squared Plus three x hedge plus. When you do a hedge by which we will get one. So this is a simplified form of the rational uh caution now that we have to approach, we have to let Hedge approaches zero, so therefore if you do this, This will become, this quantity will become zero and this will become three X square where I don't discuss it is and this quantity we have ahead, so this will become zero and finally we have one here. So therefore this is equal to three X squared plus one. So there's a little bit of a four X.

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