Topic 2: The Derivative 1 The Derivative The derivative of a function represents its instantaneous rate of change at any point along its domain. There are several ways which we can represent a derivative, with the two main methods being Leibniz notation and Lagrange notation (also known as prime notation): Leibniz notation: If y = f(x), then the derivative of y with respect to x is written as " " = " ff(xx) Prime notation: If y=f(x), then the derivative of y with respect to x is written as yy = ff xx In the previous chapter, we used the limit definition of the derivative and the alternate limit definition of the derivative to compute the derivatives of functions: Limit Definition of the Derivative: (Used to directly compute derivative) ff xx + h ff xx ff xx = lim h Alternate Limit Definition of the Derivative: (Used to directly compute derivative at x=a) ff xx ff aa ff aa = lim xx aa However, this segment of the course focuses on methods to shorten the amount of time it takes us to compute the derivative of a function. Below are the properties/rules of differentiation: 1. 2. 3. 4. 5. " cc ff(xx) = cc ff(xx) " Constant- Multiple Rule " ff xx ± gg xx = ff xx ± gg(xx) " " Sum/Difference Rule " ff xx gg xx = ff xx gg xx + ff xx gg (xx) Product Rule = " Quotient Rule " = ff gg xx gg (xx) Chain Rule (Composition of Functions) Below are more specific rules (depending on the given function) 1. 2. 3. " cc, wwheeeeee cc iiii aa cccccccccccccccc, = 0 a. The derivative (rate of change) of a constant function is always zero. This is because y=c would be a horizontal line (like y=1, y=ππ, y=150, etc..) " xx = nnxx a. This is the Power Rule for differentiation. n is some integer log " xx = "# a. Logarithmic Differentiation. aa represents the base of the logarithm. b. Because llllll = 1, llllll = = (e is the base of the natural logarithm) " "#$
Topic 2: The Derivative 2 4. i. llllxx = " " aa = aa llllll a. Exponential Derivatives. aa represents the base of the exponential function. b. Because llllll = 1, " ee = ee llllll = ee i. " ee = ee 5. Trigonometric Derivatives: a. b. c. d. e. " " ssssssss = cccccccc cccccccc = ssssssss " ttttnnnn = sec xx " " cccccccc = cccccccc cccccccc ssssssss = ssssssss tttttttt f. cccccccc = " csc xx 6. Derivatives of Inverse Trigonometric Functions: (Can be written as aaaaaaaaaaaaaa or sin xx) a. aaaaaaaaaaaaaa = " b. aaaaaaaaaaaaaa = " c. aaaaaaaaaaaaaa = " d. aaaaaaaaaaaaaa = " e. aaaaaaaaaaaaaa = " f. aaaaaaaaaaaaaa = " See examples worked out at DuranLearning.com
Topic 2: The Derivative 3 Working Backwards from the Limit Definition We ll see as we progress throughout the course that working backwards is just as important as working forward. On that note, we ll need to pay attention to some sneaky line of questioning we may see. For example: Compute lim What we should realize is that we no longer need to compute this limit directly if we understand that this limit represents a derivative. This will make the difference between solving this limit in 5 seconds as opposed to 5 minutes. Here s what we mean: Limit Definition of the Derivative Our Example ff xx = lim ff(xx + h) ff(xx) h lim xx + h xx h We are computing the derivative of ff(xx) We are computing the derivative of xx With this observation, we can use the power rule to note that " xx = 3xx, and thus lim xx + h xx = 3xx h Seems convenient, right? Take a note of this. Most instructors will challenge your ability to recognize the limit definition of the derivative when given. You may not always be taking the limit of h either. We can see variations such as lim xx + qq xx qq Don t be fooled. This is the exact same limit and derivative computation. See more examples of recognizing the limit definition of the derivative at DuranLearning.com
Topic 2: The Derivative 4 Differentiability We will see this term thrown around throughout the course. A function is differentiable if its derivative exists at each point in its domain. This implies that a differentiable function is also a continuous one. In its most basic explanation, a function is differentiable if: 1. It is continuous on its domain and 2. Its derivative is continuous on its domain. We will often see problems which ask us to determine the value of constant which will allow a function to be differentiable. Below is an example: xx Determine the value of aa which makes ff xx = {, aaaa + bb, xx < 5 xx 5 differentiable. The best way to approach such problems is to first determine the relationship between aa, bb and our function which will guarantee ff xx to be continuous at xx = 5. Afterwards, we should create our relationship which will make ff(xx) continuous at xx = 5 because we will often solve for the value of one of our unknown constants easily. Here is what we mean: For ff(xx) to be differentiable, ff xx must be continuous at x=5. Therefore, " xx = " (aaaa + bb) when xx = 5 2xx = aa when xx = 5 (note: aa and bb are constants, and so aaaa = aa and bb = 0) " " 2 5 = 10 = aa We have determined the value of aa, which we can use to determine the value of b. Remember, differentiability implies continuity, and so ff(xx) must be continuous at xx = 5: xx = aaaa + bb at xx = 5 5 = 10 5 + bb Substitute known values for xx and aa 25 = 50 + bb 25 = bb Solve for bb Therefore, the values of aa and bb which make ff(xx) differentiable are aa = 10 and bb = 25. Note: We can certainly start by first forming the relationship to make ff(xx) continuous before making ff (xx) continuous, but I chose the opposite order because it allows us to immediately solve for one of our unknown values. See more examples of differentiability at DuranLearning.com
Topic 2: The Derivative 5 Emphasis on the Chain Rule Out of all our rules and properties of derivatives, the chain rule is the toughest one; not because it is more complicated, but because the majority of students don t understand compositions of functions fully, which is what the chain rule stems from. If we master compositions of functions, we can compute virtually any derivative that involves the chain rule in one step, with proper practice of course. First, let s repeat the chain rule: " ff gg xx = ff gg xx gg (xx) Now let s repeat the chain rule with a redundant step: " ff gg xx = ff gg xx gg xx It is easy to overlook including " " because = 1, but that s the point: Continue applying the chain rule " " until you receive as that is the indicator that we have completed the chain rule process. Take the following example, for instance: Compute " if yy = xx " By using the power rule, we would receive: " = 2xx, which is correct, but let s take a moment to " consider this same problem in a different light: Compute " if yy = xx " We can view xx as a composition of functions. If ff gg xx = xx, then we can say ff = For now, let s fill in the blank with something ff sssssssssshiiiiii = sssssssssshiiiiii By computing the derivative of both sides, we ll now have: ff sssssssssshiiiiii = 2 sssssssssshiiiiii (sssssssssshiiiiii) If we simply replace that something back with xx: since " " = 1, ff xx = 2 xx ff xx = 2xx
Topic 2: The Derivative 6 This is a very common example where the shown process is redundant because we can simply use the power rule and move on. However, understanding that the chain rule applies to this and all derivatives is essential to mastering the chain rule and implicit differentiation. Watch, Practice, and Master this concept with our videos on the chain rule See more examples of the Chain Rule at DuranLearning.com
Topic 2: The Derivative 7 The Chain Rule and Implicit Differentiation The chain rule is easily the most misused rule of differentiation. We need to make sure we understand and master the chain rule before moving into implicit differentiation: First, we need to understand the difference between " For example, we need to understand that " " " = 1 aaaaaa 1. = " " " " " We next need to understand that the chain rule applies to all derivatives. For example: " xx = 2xx = 2xx " aaaaaa (and the many other forms). " We re used to using the power rule for a derivative such as the one above, but the chain rule applies if we treat xx as the variable of substitution for the chain rule. How does this help? Suppose we want to compute " yy. We ll have to apply the chain rule and treat yy as a something : yy = sssssssssshiiiiii = 2 sssssssssshiiiiii (sssssssssshiiiiii) Translates to: yy = 2yy We need to always pay attention to the variables which we are taking the derivative of as well as what variable we are taking the derivative with respect to. In the next main topic of the course, we ll discuss the concept of related rates, where we take the derivatives of variables with respect to time, and not x. This is implicit differentiation applied to the real world. What is implicit differentiation? - Don t worry. Implicit differentiation simply means we are computing a derivative of an equation where it s not of the usual form yy = ff(xx). Most times, we ll have an equation where we can t easily solve for y to attain the form yy = ff(xx) before taking the derivative, like ssssssss + yy = 2xx. However, we can still solve for " quite easily if we have a strong understanding of the chain " rule. Let s look at the following example regarding implicit differentiation:
Topic 2: The Derivative 8 Compute " where xx + yy = 10 We will need to apply the chain rule carefully [ xx + yy = 10 ] xx + yy = 10 3xx + 2yy = 0 3xx + 2yy " " = 0 (note that " " = 1 bbbbbb 1) " " From here, we will need to solve for ", which will take some common algebraic skills " Subtract 3xx from both sides 2yy = 3xx Next, we will divide both sides by 2yy, thus giving us = 3xx 2yy And that s it Implicit differentiation involves two main steps. When solving for " ", 1. Take of both sides of the equation " 2. Algebraically solve for " " Yes, our answer can have y and x in it. **Note: and yy are not the same variable. If we understand the chain rule before tackling implicit differentiation, the process can seem very easy. Before we attempt to tackle Topic 3: Applying the Derivative, we must make it a point to master the chain rule, which translates to mastering implicit differentiation, which will allow you to master Related Rates which is arguably the hardest concept in the first half of Calculus I.
Topic 2: The Derivative 9 See more examples of Implicit Differentiation at DuranLearning.com