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(a) If $ f(x) = x^4 + 2x $, find $ f'(x) $.

(b) Check to see that your answer to part (a) is reasonable by comparing the graphs of $ f $ and $ f' $.

a) $4 x^{3}+2$

b) Notice that $f^{\prime}(x)=0$ when $f$ has a horizontal tangent, $f^{\prime}(x)$ is positive when the tangents have positive slope, and $f^{\prime}(x)$ is negative when the tangents have negative slope.

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This is problem number thirty three of this tour. Calculus. Eighth edition, Section two point eight. Party if ever Vex equals X to the fourth plus two. Ex find every crime of X king. Let's use the Internet derivative definition. Find the paramedics limit as h approaches zero of this function of X. You know, you wanted a explosive JJ to That means explosive quantity to the fourth power. Plus two times actress H they were gonna subtract the function of X X to the fourth US to X is all divided by just each. Okay, good. Next step is to expand every term. Here we have the binomial to the fourth power. Um and then she read some other numbers as well. So this binomial to the fourth power will be X to the fourth. Because for X cubed H plus six x squared each squared plus for X h cute plus each to the fourth. That's just this final meal to the fourth power. And we're going to distribute the to here to the X and age plus two x plus to age. And they were in a subject exit. The fourth and two exits one okay, and this is all over H I'LL take a look at the numerator and see what we can cancel out a positive excellent forth and a negative x a fourth to those go away a positive to X and negative tricks and then an agent A dominator will cancel with an ancient each of the terms in the numerator since each of the terms of numerator have at least one each king. So what we're left with is the limit as h purchase zero of for X cubed less six ax squared each plus for eggs each square pas age cubed plus two Now, as a chair purchase zero each of these age terms a purchase zeros for those No way and we should be love with for X cubed plus two And this will indeed be our derivative of paramedics. Now we will verify and party by chicken or answer with party um to make sure that it's reasonable comparing the graph of f end of crimes were going plot both FX excellent forthwith to X And if Prem of X for Cuba for X cube two plus two and see if it is, this answer is reasonable. So we have pulled up at the next, which is in blue, and the derivative, which is in green here with their functions that we were given in which we found. And let's discuss whether this makes it, since the function F of X is a something similar to a problem. So on the left side it's decreasing, and when it's decreasing, its slope is negative, so as a very large negative slope. But then it gets less negative as it gets close to this minimum point. So that's what we see here. The function, the green function, the dirt of function is mostly negative. Insolent reaches is until it keeps approaching Weikel zero. So purchase of slope of Zero exactly where the minimum is. So this is consistent, the slope of zero at the minimum, so that is correct afterward, of the soap increases, the slip is positive for the remaining part, and this is shown on the slope craft, the derivative crafting green and what we see here is that the slope increases and then stays the same for a little bit and then increases again. And that's what explains his behavior here. This soap is positive initially and increases and becomes more positive until a point here. Actually, the slip seems a little constant exactly equal to around two. So the slopes ese constant for a little bit. But then it starts increasing again. Until this derivative that we found Brinkley is consistent with our two cafs shown here.