Please solve A-E

1) A Guided Derivative Calculation The derivative shortcut formulas allow us to take derivatives of very complicated functions. However, it can be difficult to determine exactly how these rules work together when taking the derivative of a complicated function. This problem will walk you through a process breaking down the derivative part-by-part using logarithmic differentiation. Each part will only require one of the derivative rules we have learned. but in the end we will have calculated the derivative of the function: {tanth (3 + 5")\" 'lsin_1(x)+4x Notice that this fraction has three factors being raised to powers. multiplied. and divided together. To simplify our notation, we will make up some new variables: y. u. v. and w. to stand for those factors as follows: f(x) = y: {(x}, u=tan(x), v=3+5", w=sin'1(x)+4x a) We will need the derivative of each of these factors. The derivatives of u. v. and w will be denoted u', v', and w', respectvely. Find the derivatives of those Factors individually. Your answers should only use the variable x. J' i) (4 points) it = di;(tan(x)) = .. . (71 X 7 u) (4 pomts) v dx (3+5 ) iii) (4 points) w' = d1; (sin1(x) + 4x) = b) Now we will use these simpler y, u, v, and w in our calculation to stand in for the more complicated factors. First we write our formula in terms of y, u, v, and w: \"v\"? y:vx'/- Using logarithmic differentiation. we start by taking the logarithms of both sides of this formula then applying the properties of logarithms. x/T U | = | W) n (w W) = In (u') ln(v"} In (W) =ln(u)xln(v)-%lniwl Let's give names to these individual terms by setting 1 a=|n(y), b=\\/Tln(u), c=xln(v). d=ln(w). That makes this formula simplify as a : b i c i d. The derivatives of a, b, c, and d will be denoted by a\c) We know a\" = b' c' d'. Substitute what you found For a' in part b)i), and solve for y'. Note: Your answer here should only include the variables y. b'. c'. and a\". a'=b'-c'd', so i) (4 points) 3/ = d) In parts b)ii), b)iii), and b)iv), you Found Formulas For 13', c', and d'. Substitute those Formulas into 13', c", and d' into your answer in c)i) to find a formula For y' in terms of the variables y, u, u', v, v', w, and w'. i} (4 points) y" = e) Our final step is rewriting the y, u. u', v, v'. w. and w', in terms of the single variable x. Since y : x). this means we are finally finding the derivative we started with. Using your formula for y' = f'(x) from part d)i) substitute each copy of y. u. u', v, v', w, and w', in terms of only the variables x using the formulas you found in part a). Note: Your answer for this part should only contain the variable x. i) (4 points) f'(x) =