find-the-derivatives-of-the-given-functions-r05leftsin-1-3-tright4-4

Question

Find the derivatives of the given functions.
$$r=0.5\left(\sin ^{-1} 3 t\right)^{4}$$

Tyler Moulton · Accepted Answer

we want to find do I. D. T. For the expression why is equal to 0.5 times the arc sine of three T raised to the fourth power. Just all this problem, we&#x27;re going to rely on derivative shortcuts we picked up throughout single variable calculus in addition to the inverse trig derivatives that we&#x27;ve recently learned. So I&#x27;ve listed here the major inverse trig derivatives were going to primarily rely on the fact that D. D. X. Marks an active route 1 1/1 minus X square. We&#x27;re also gonna need to use the chain rule since our arc sine is raised to the power of four and has argument three T. We note that why is already written in the most easily differential forms we perceive solved by the chain rule we have DY DT equals 0.5 times four times are three T. Cute times the rhythm of what&#x27;s inside which is three over route one minus 90 squared, Multiplying out our constants and simplifying the expression gives DY DT equals six arc sine three T. Cute over route one minus 90 squared

Robert Daugherty · Answer

Section three. That six problem 25 were dealing with derivatives that involved the chain rule. So I have a function s is a function of t to find its derivative. So we break it down piece by piece. There is a constant for over three pie. What is the derivative of the sine function? That&#x27;s the coastline function to this is co sign of three tee times. What is the derivative of three t? That is three plus 4/5 pi. The derivative of the coastline function is the minus. Sign five t the derivative of five t is just five. And so now they&#x27;re since, um, pimp it simplification with these coefficients. And so your final answer becomes four over Pie co sign of three T minus four over pie sign five t You could factor out the four over pie in that case as well. So that is the derivative of this function s as a function of t

Amy Jiang · Answer

Okay, since we have the product of two variables, let&#x27;s let the following you on that. This we ve And then let&#x27;s use our product role could take the derivative. Okay, so have you, Prime. That&#x27;s four times we and thats plus b prime. So we&#x27;re gonna use General on this who have four times two t one of our 543 truecar four minus one. That&#x27;s three. And then we need to take the derivative of the inside as well. That&#x27;s times 10 x 22 power for and then we want to fight that by we have 40.

Robert Daugherty · Answer

Section three at six. Problem number 44. We&#x27;re dealing with the chain rule. One of the first things I would do with its function. If this is gonna be easier to think about it, if I take care of that negative exponents. So right, this as three monies to t divided by one plus the sign of three t. Now I can use my question rule. So I know that G prime of tea is going to be the derivative of the numerator, which is minus two times of the denominator minus the derivative of the denominator, which is going to speak co sign of three t times three and then times the numerator all of this over that denominator squared one plus sign of three t quantity squared. And so if you look at this, um, what we come up with in this case, IHS thing is equal to, um you&#x27;ve got minus two times one plus side of three t. And then it was like I&#x27;ve got minus three three minus two t co sign of three t All of this over one plus sign of three T Square. There&#x27;s really not much simplification you can do in. Ah, in this particular case, eso That should be our final answer. No, you could do. I mean, somebody could look at whether the minus sign this in common so you could play around and get different variations off that so you could end up with I could take care of distributing this out, but really, there&#x27;s no need here, So this should be this close to the final answer is you&#x27;re going to get without any further simplification.

Jonathon Brumley · Answer

the derivative of a constant to a you power where you it is something that contains an axe is the constant to the U power times, the derivative of that You times the natural log of our base. So if we had something like five to the three t power, it&#x27;s derivative would be five to the three t power times, the derivative of the U and the derivative of three tea with respected T is three and then we&#x27;ve got the natural log of the base of five. Now, if we want to, we could put this three up on the power of five by natural law rule and five to the third is 125. So natural log 125 times five to the three T power.

Amy Jiang · Answer

Okay, So before we take our derivatives, let&#x27;s start by you writing it in terms of, you know, united. As I recall, if we have p of X, we can rewrite that even her of X a wannabe. So we can rewrite our term as let&#x27;s see, our base is sign of use. Do we have you to the power of our X is the square root of tea that&#x27;s also teach a part of 1/2. But didn&#x27;t we have Ellen of finance? Okay, I don&#x27;t Let&#x27;s take our David. So, um, you have you have this evening and then we want to take the derivative over inside, so we&#x27;re gonna use our product well, here. So let&#x27;s start with our derivative for, um t. So that&#x27;s 1/2 t to the power of 1/2 minus one. That&#x27;s negative. One and then times are meeting term. And then plus the derivative about one that&#x27;s one over sine of t time did derivative of fine, which is co site. So we have coastline over sign. So that&#x27;s equal to uncle change in o t. And then times the remaining which is t to the power of 1/2. Okay, so we get the following for our derivative

Problem

Find the derivatives of the given functions. $$y=…

Problem 25 Easy Difficulty

Find the derivatives of the given functions.
$$r=0.5\left(\sin ^{-1} 3 t\right)^{4}$$

Answer

$\frac{d u}{d t}=\frac{8 \sin ^{-1}(4 t+3)}{\sqrt{1-(4 t+3)^{2}}}$

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Allyn J. Washington, Richard S. Evans

Basic Technical Mathematics with Calculus

Catherine R.

Heather Z.

Caleb E.

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Precalculus Review - Intro

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$\frac{d u}{d t}=\frac{8 \sin ^{-1}(4 t+3)}{\sqrt{1-(4 t+3)^{2}}}$

Basic Technical Mathematics with Calculus

Catherine R.

Heather Z.

Caleb E.

Joseph L.

Precalculus Review - Intro

Functions on the Real Line - Overview

Allyn J. Washington, Richard S. EvansBasic Technical Mathematics with Calculus

Catherine R.

Heather Z.

Caleb E.

Joseph L.

Allyn J. Washington, Richard S. Evans

Basic Technical Mathematics with Calculus