Worksheet: More practice with the idea of "implicit" When we say "differentiate with respect to x, that means that we are treating x as the independent variable. (i We indicate this when we use 3 1. Find % (axz + sin(bx) c) is the independent variable, and we can say that we are I} If we use jt, that indicates that "differentiating with respect to - i 3 t 2.Flnd at" +2) We can really treat any variable as the independent variable; the notation tells us what letter to consider the independent variable. 3. Find % (7 ln(y) + yarctanO'D Often, we use x to represent the independent variable, and y to represent the dependent variable. In other words, we think of y depending on 1'. Given an equation, we can plug in values of x and find the corresponding values of y. 4. Given y = 3x + 4, find 37 when x = 1. Sometimes, when we plug in our x value, we may have to do a little more work to find y. 5. Given 9x2 + 4312 = 25, find 3/ when x = 1. If we want to di'erentr'ote with respect to x, that means we want to find how 3! changes when we change x. More precisely, we want to find the rate ofchange ofy with respect to x. To do this, we are going to take the derivative of both sides of whatever given equation; Example: Given an explicitly defined relation between at and 3!, like y = 3x + 4: d (1 50') = 3835+ 4) LEI35(3)) is just the derivative of y with respect to x, which we notate by :: or y'(x) or just 3!". If we have an implicitly defined relation between x and y, we are starting off with the same idea, but we have to remember that our independent variable is x, and y is actually dependent on x; that is, y = y(x). 6. Find -, given that 9x2 + 4y2 = 25. We remember that since y = y(x), that is, y depends on x, or y is a function of x, we sometimes have to use the chain rule! If y is inside of any other function, that means we actually have a function inside of a function. 7. Given y + sin(xy) += = ex, find dx ( ) + sin(xy ) + =) = (ex)Similar ideas apply when we are doing related rates problems. In related rates problems, we are usually looking at how different quantities are changing with time. In other words, we are treating time t as the independent variable, and any other quantities that are changing with time as dependent variables. Example: Suppose a snowball is melting. In this case, as time goes on, both the radius and the volume are changing. Suppose we want to find out about the rate of change of the volume, with respect to time. This means, we want to differentiate with respect to t. The volume of the snowball is given by V : Ems. We can start by taking the derivative with respect to t on both sides ofthis equation: 01 d 4 _ :_ _ 3 arm dt(3m) BUT, we need to remember that actually, V changes with time, so V : V(t); and also r changes with time, so it" : r0"). % (11(0) is just Va) When we differentiate the right hand side, 61m"), we will need to use the chain rule d d 4 3 50"") : gem") ) v'm : 4n(r(t))2r'(t) 8. Find -, given that A = xy, and both x and y are changing with time. 9. Find -, given that y In(x) + Vy = cos (x) ex . (Note: the notation tells us that y depends on x, that is, y = y(x). 10. Find a, given that r2 - 3y = 2. (Note: in this case, y is the independent variable! And r = r(y) is the dependent variable.)