Math 1151, Autumn 2023 Written Homework 3 Page 2 of 5 Problem 1: Guided Derivative Calculation (38 points) One of the joys of the Derivative Rules we have learned is how they allow us to take derivatives of very complicated functions. However, it can be difficult to figure out exactly how these rules work together when taking a complicated derivative. This problem will walk you through a process of breaking down the derivative partbypart using logarithmic differentiation. Each part will only require one of the derivative rules we have learned, but in the end we will have found the derivative of the function: (3XT1'l' 5X)12 - (2X5 + 3 ln(X))SeC[X) g(X) _ (l tan1(X))5X - (2x)5l"l2xl Notice that this fraction has many factors being raised to powers, multiplied, and divided together. Each term that is being multiplied is called a multiplicative factor. a) (8 points) This is a lot of writing here, and the complicated pieces makes it hard to analyze the entire function. To make this simpler, we make up some new variables, u, v, W, z, to stand for the base of each multiplicative factor as follows: u : 3x'1 + 5x, V : 2x5 + 3|n(x), w _ 1tan'1(x), z _ 2x We will need the derivative of each of these individual terms. The derivatives of u, v, w. and 2 will be denoted u/, V/, w', and 2', respectively. Find these derivatives using your shortcut rules, and fill in the blanks below. Your answers for this part should only use the variable x. d , u':$(3x 1-|~5X) : v' : 1(2x5 + 3|n(x)) : dx w' : 1(l i tan'1(x)) dx 2' : i(2X) : Math 1151, Autumn 2023 Written Homework 3 Page 3 0f 5 b) (20 points) Now we can use the simpler u, v, w, and 2 terms to make our calculations easier to understand. First we write our function in terms of u, v, W, and z: U12 ' Vsec(X) g(X) : st . Zsin(2x)' Isn't that easier to read? The first step in using logarithmic differentiation is to rewrite our function so its base is the constant e. This step naturally brings in a logarithm, which we use in the next step: g(x) : e|n(g(X))_ Now we can focus on the exponent, using the logarithm to simplify the complicated fraction: \"12 . Vsec(x) |n(g(x)) : In (W) : In (\"12) 'I' In (Vsec(x)) _ In (W5X> _ In (25in(2x)) 12|n (u) + sec(x) In (v) 7 5x In (w) i sin(2x) In (2) Let's give names to these individual terms by setting: a : 12In(u), b : sec(x)ln(v), c : 5x|n(w), d : sin(2x)ln(z)t This notation simplifies our equation to: In(g(x)) : a+ b c d. l i We will need the derivative of each of these individual terms. The derivatives of a, b, c, and 0' will be denoted by a', b', c and (1/ respectively Find these derivatives using your shortcut rules, and fill in the blanks below. Your answers for this part will contain the variables u, u', v, v', w, w', z, z' and x. Do not substitute or write answers only in terms of x. , d a : dX(12|n(u)) : If d 7 b &(sec(x)ln(v)) , d c : &(5XIH(W)) : d' : i(sin(2x) In(z)) : Math 1151, Autumn 2023 Written Homework 3 Page 1 of 5 We have done the hard work of breaking up the function into pieces, and nding the derivatives of each piece. Now we will put these pieces back together in the right order to find the derivative of the original function. We'll do this in a few steps. c) (4 points) Using our new variables, we have: g(X) : ea+b7c~d- Take the derivative of this equation to find an expression for g'(x). Write your answer in the space provided below. Your answer here should only include the variables a, a', b, b', c, c', d and d'. g/(X) d) (3 points) In Part b you found formulas for 3/, b/, c', and d'. We also know that eakc'd : g(x). Substitute those formulas into your answer in Part c to find a formula for g'(x) using only the variables g(x), u, u', v, v', w, W', and x. Your answer here should not include any of the variables a, a', b, b', c, c/, d, or d', or the constant e. g/(X) e) (3 points) Our final step is rewiting the variables g(x), u, u', v, v', w, w', z and z' in terms of the single variable x. Using your answer from Part d, substitute each copy of u, u', v, v', W, and w' in terms of only the variable x using the formulas you found in Part a. You should also substitute out g(x) for our original function. Your answer for this part should only contain the variable x. Your answer will be very long. You may want to neatly use two lines to write the full expression. (continued)