1)I f()=(2x+3)(+' 3.+7) fird f "(x)12)1 f(x) = (2x)x' +I)(x-3) find f %(x) (}First expand the functioni und then apply differentizticn mle: (6) Tken ...

$1-6$ Write the composite function in the form $f(g(x)) .$ $[$ IIdentify the inner function $u=g(x)$ and the outer function $y=f(u) . ]$ Then find the derivative $d y / d x$
$$y=\left(2 x^{3}+5\right)^{4}$$

Okay. This question wants us to find the derivative using the chain rule. So to start, we should notice that this to execute plus five function is trapped inside that fourth power. So our inter function G of X is two x cubed plus five. So now we can write this as why is equal to G of X, all raised to the fourth. And then we confined this derivative using the chain rule. So de y dx would be the derivative of the outside leaving the inside alone, then times the derivative of the inside. So now we just need to plug in for everything. So g of X, we said, was two x cubed plus five. So the derivative of that would just be six x squared and then simplifying we get D y DX is 24 x squared times two x cubed plus five, all raised to the third power

Question six people of fine this because the writers in the form for composite function where there will be one dinner and one out of function is there enough function is clearly this so enough function as Tu minus areas to the poor ex. And if you talk about the function, that will be the one square root so the outer function will be square root human. So now if you're gonna find the differentiation So we have to use change rule over here when we differentiate tu minus areas to the bod X rays to the bargain over to so one over to comes down. We have two minus areas to the bod X rays to the ball minus 12 minus one or two. And then we differentiate the inside expression which is two minus areas to the products. This can be read it and has won over two square root off to minus ages to the vortex on here that the transition will be zero minus areas to the barracks. So the final answer consort is negative. It is to the products over two squirrels off to minus it is to the products. So this is the required answer

Question one. In this case, we have to write this function as a composer. Function off the form F off GXE. The F is the outer function and G Is this function eso? Let's talk about if which is drafted as a few. So if if you read a dysfunction that can be read it on a compass four X rays to the ball, one tree so clearly the inside function is the so that G X on the outside function will be, uh, can build on a US to the bar one or three that is the outside function. So let's denoted by if you there's all the function and the inside function will be one plus four x So we'll say that inside function, which is G X, is equal to one plus four x so that the F off G X would be there's now in order to differentiate this, we gotta use chain rule over here. So Di Vaio over DX would be, uh, will say f dash you times Uh uh over here will have its better to write it as d or d x off F o g X differentiation off F g X would be f dash G X names Judi XIX. Now, the differentiation off footage, uh, is, uh, in fact, this we go to differentiate the power of us. One of the three will come down one plus four x rays to the bar, minus 2/3 on the differentiation of one plus four X is just four. So the answer. Consult us for over three times. This is a power and negative, so we'll take it down. So that can be written as one plus four X rays to the power to three. So this is the required. And so for the B Y over here.

Question five eso In this case, we gotta find a composer function where they will be one dinner and one outer, clearly the inner estes. So if you take GX s route text the now trouble definitely be the one that e. So the outer function will be here is to the power you And when we differentiate this people who use the change rule over here, the difference station off E S e power remains other players, and then we differentiate the power, which is Tex. So this concert as it is to the power techs and the condition of Texas one over to text. So the final answer consort as it is to the power route, takes over to rule Tex. So this is the final answer.