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 ```When an equation is differentiated we are finding an equation which shall enable us to calculate the rate of change on the graph of that equation at any point we may chose. In many situations it is important to know the rate of change at a chosen point on a graph of a function. for a a graph of a function which represents a staight line, it is quite simple to identify the grident of that line either geomettrically or algebralically. If a graph of a function is a curve, the graident is different at different points on the curve. It is therfore much more difficult to calculate the rate of change at any point on the curve. This is why we use the rules of differentiation to minipulate a given function and therfore enable ourselves to calculatethe rate of change at any point. 1/ examples differentiate y=(2x+3)^2 let u = 2x+3 then du/dx =2 and y=u^2 then dy/du = 2u since dy/dx = dy/du * du/dx by substituting dy/dx = 2u *2 = 4(2x+3) 2/ y= (x^2 -3x+5)^3 let u = x^2 - 3x+5 du/dx = 2x-3 y = u^3 dy/du = 3u^2 again substituting dy/dx = dy/du * du/dx = 3u^2 8 (2x-3) = 3(x^2 - 3x +5)^2 (2x - 3) --------------------------------------------------------------------------- answer 4 questions show all working and simplify as much as possible giving reasons for answers. differentiate the following 4 questions 1/ y = (x^2 +2)(3x - 1) 2/ y = (2x - 4)^2 3/ i = 20 sin (100 pi t +pi/3) 4/ i = 50 cos (50 pi t - pi/6)```
 ```Thank you for the questions. Here are your answers. Question 1 ---------- We differentiate y= (x^2+2)(3x-1) with respect to x . We will expand the function into a polynomial and the differentiate that polynomial: y=(x^2+2)(3x-1) = (x^2+2)3x + (x^2+2)(-1) = 3x^3+6x -x^2 -2 = 3x^3 -x^2 +6x -2 Now can just differentiate this polynomial directly: dy/dx = 9x^2 - 2x + 6 This is the answer. Question 2 --------- In this question and the succeeding questions, we have to find an appropriate u so that we can use the chain rule to simplify the computation. We differentiate y= (2x-4)^2 with respect to x . We set u=2x-4 . Then y=u^2 . Hence, dy/du = d(u^2)/du = 2u and du/dx = d(2x-4)/dx = 2 . Hence, by the chain rule: dy/dx = (dy/du) (du/dx) = 2u (2) = 4u = 4(2x-4) = 8x-16 Question 3 ---------- We differentiate i=20 sin (100 pi t + pi/3) with respect to t. We let u= 100 pi t + pi/3 . Then du/dt = 100 pi . We have i=20 sin (u) so di/du = 20 cos (u) . Hence by the chain rule, di/dt = (di/du) (du/dt) = 20 cos(u) 100 pi = 2000 cos(u) = 2000 cos (100 pi t + pi/3) Question 4 ---------- We differentiate i = 50 cos (50 pi t - pi/6) with respect to t. We set u= 50 pi t - pi/6 . Then du/dt = 50 pi We have i = 50 cos (u) so di/du = -50 sin (u) Hence by the chain rule: di/dt= (di/du)(du/dt) = -50 sin (u) 50 pi = -2500 sin (u) = -2500 sin (50 pi t - pi/6) ====================== Plots: I have made simple plots of each function and its derivative. This might help you to visualize what is going on with the functions and the derivatives. One simple thing to observe in these plots is that whenever the function has a local maximum or minimum (that is, at the top of a "hill" or the bottom of a "valley" in the graph of the function) the value of the deriative is always 0 there. This is a general property of the deriviative - it's one of the important uses of deriviatives, actually, as it is used in physics in a lot. The plots are at: http://www.rbnn.com/google/differentiation/1.jpg http://www.rbnn.com/google/differentiation/2.jpg http://www.rbnn.com/google/differentiation/3.jpg http://www.rbnn.com/google/differentiation/4.jpg for questions 1 through 4 respectively. I used Matlab to generate these plots. Matlab is an excellent program, see http://www.mathworks.com , and I highly recommend it for playing around with mathematical functions. ================== As always, if you have any questions at all or would like additional information, please use the "Request Clarification" button to solicit more information before rating the answer.```