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Question

Find the derivatives of the given functions.
$$y=3 \sin ^{3}\left(2 x^{4}+1\right)$$

Tyler Moulton · Accepted Answer

uh huh. We want to find dy dx for Y is equal to three kind of sine cubed, two extra, fourth plus one to do so we rely on derivative shortcuts we picked up throughout single calculus that is single variable calculus excuse me. Combined with specifically trigger metric derivatives. So first we have relevant rules listed here on the last zero DDX. In Mexico Cossacks. DDX Cossacks is negative syntax. We also have rules one through three listed where one is the power rule to the product rule through the chain rule. In this particular problem we&#x27;re gonna have to use rule +01 and three to solve. So first we write our derivative in a way that&#x27;s easy to solve for. So the function Y equals three signed two X to the fourth plus one cubed. This makes it easy to use the train rule. Thus we have dy dx is three times to sign square. Two extra. The fourth plus one times the derivative was inside which is co signed to actually the fourth plus one times eight X cube. Giving a final solution. DY Dx equals 48 X cube. Sine squared two extra. The fourth plus one, co signed to actually the fourth plus one.

Shumayal N · Answer

for more complicated functions like these, you start with the derivative of the outermost function and you work your way in. So the derivative of sign, which is the outermost function, is cool. Sign and you write the inter function as it ISS for X cubed plus three eggs, most one. And then you take the derivative of Thean Er. Function separately so we&#x27;ll do D over the X or X cubed plus three eggs plus one. Now the derivative of four X cubed is going to give us four into three X squared. The derivative of three eggs will give us three into one and the derivative of 10 Hang. Do you already have call sign or ex cute? I love three eggs plus one, which give this our final answer of love. X squared plus three well supplied room Total sign oh or execute plus three x plus one

Philip Christensen · Answer

find the derivative of the function F of X, where X is equal to two all raised, or to race to the power of the quantity. Three plus sine X. Now we need to reboot one room off anything raised to the X exponential functions that he derivative anything of the form because of a to the to a function g of X, it&#x27;s equal to a to the G of x times, the natural walk of a then using the General Times cheap crime, his ex. So I can use this to solve for F prime. His acts are derivative of f of X, so we have that form. So we have 2 to 3 plus sign backs. That&#x27;s our G of X times natural log to then we need to take the derivative of G of X Year of axe sign packs. So our G prime of X is equal to three is a constant, so that goes to zero zero. Plus, he derivative of sign of X is co sign X. So then we multiply this by co sign hex that see derivative halfbacks

Heather Zimmers · Answer

Here&#x27;s a composite function that we&#x27;re going to differentiate using the chain rule and the good news is that even though there are going to be a lot of steps because we have a lot of layers in this composite function, there&#x27;s not going to be any simplifying. Weaken do and we&#x27;re done. So that is often a relief because the simplifying is usually the most complicated part of the process. So let&#x27;s find the derivative notice that the outermost function is the fourth power function. So we bring down the four and we raised the inside to the third. So we just write the entire inside over again. But now it&#x27;s to the third. Now we multiply by the derivative of the inside. So just focusing on this part, which is a some and the derivative of the sum, is going to be the some of the derivatives. So the derivative of X is one plus. Now we&#x27;re finding the derivative of this inner part, and we&#x27;re going to need to use the chain rule on that. So we bring down the three and we raised the inside to the second. Then we multiply by the derivative of the inside, and the inside is a some. This part here is a some and so it&#x27;s derivative will be the some of the derivatives. The derivative of X is one plus. Now we need the derivative of sine squared X, and remember that sine squared X means the sign of X quantity squared. And so it&#x27;s derivative would be, according to the chain rule, two times a sign of ex because you bring down the two and you raise sign to the first multiplied by co sign of X because that&#x27;s the derivative of sign. So that&#x27;s what we have next to sign X Co sign X. Okay, we made it to the very inside and then we look at that for a minute and we ask ourselves, What can we combine? What can we factor out? And we realized that because of the plus signs, primarily it&#x27;s kind of stuck the way it is. So there you have it

Sid Wan · Answer

number twenty six will define the derivative off. Why Wass Science three acts plus Coast Coat Coat, X coat hand execute with Ace So first we take this as a oh ho ho. It&#x27;s eight off this part. It&#x27;s Cube servants says this power, and we have to multiply derivative off. What&#x27;s inside? So your own people first, this part is going to be co signs for excellent blustery channel Klaus coat hands. Ex Directive is and collective Call C can&#x27;t Max Cube Square multiply derivative of Tex Cube. So relax squared channel. So even assemble. Devise a signs of post co tenant execute Sellards by three, cause I Thrax liars throughout square Cosi because he can&#x27;t execute square. Yeah, that&#x27;s it. Just used power and general inside work for this one.

Todd Vawdrey · Answer

Okay, so this problem we&#x27;re taking the derivative of y equals negative three e to the to sign X. So this is a composition of functions or have this function right here inside of a negative three e to the X function. So the derivative using the chain rule is just going to be the outside funk, the derivative of the outside function, which is three e to the to sign X times the derivative of that green function which is to co sign X so all together we can go ahead and write that as negative six e to the to sign ex co sign X oh.

Problem

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

Problem 12 Easy Difficulty

Find the derivatives of the given functions.
$$y=3 \sin ^{3}\left(2 x^{4}+1\right)$$

Answer

$\frac{d y}{d x}=-15 \cos ^{2}(5 x+2) \sin (5 x+2)$

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Allyn J. Washington, Richard S. Evans

Basic Technical Mathematics with Calculus

Grace H.

Catherine R.

Caleb E.

Michael J.

Precalculus Review - Intro

Functions on the Real Line - Overview

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$\frac{d y}{d x}=-15 \cos ^{2}(5 x+2) \sin (5 x+2)$

Basic Technical Mathematics with Calculus

Grace H.

Catherine R.

Caleb E.

Michael J.

Precalculus Review - Intro

Functions on the Real Line - Overview

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

Grace H.

Catherine R.

Caleb E.

Michael J.

Allyn J. Washington, Richard S. Evans

Basic Technical Mathematics with Calculus