Chapter 4. Section Derivatives of Exponential and Logarithmic Functions

Transcription

1 Chapter 4 Section Derivatives of Exponential and Logarithmic Functions Objectives: The student will be able to calculate the derivative of e x and of lnx. The student will be able to compute the derivatives of other logarithmic and exponential functions. The student will be able to derive and use exponential and logarithmic models. The Derivative of e x h e 1 We will use (without proof) the fact that lim 1 h 0 h We now apply the four-step process from a previous section to the exponential function. Step 1: Find f (x+h) Step 2: Find f (x+h) f (x) Step 3: Find f ( x h) f ( x) h Step 4: Find lim h 0 f ( x h) h f ( x) Result: The derivative of f (x) = e x is f (x) = e x. 1 P a g e

2 Caution: The derivative of e x is not x e x 1 The power rule cannot be used to differentiate the exponential function. The power rule applies to exponential forms x n, where the exponent is a constant and the base is a variable. In the exponential form e x, the base is a constant and the exponent is a variable. Ex: Find derivatives for f (x) = e x /2 f (x) = e x/2 f (x) = 2e x +x 2 f (x) = 7x e 2e x + e 2 2 P a g e

3 The Natural Logarithm Function ln x We summarize important facts about logarithmic functions from a previous section: Recall that the inverse of an exponential function is called a logarithmic function. For b > 0 and b 1 Logarithmic form is equivalent to Exponential form y = log b x x = b y Domain (0, ) Domain (, ) Range (, ) Range (0, ) The base we will be using is e. ln x = log e x The Derivative of ln x We are now ready to use the definition of derivative and the four step process to find a formula for the derivative of ln x. Later we will extend this formula to include log b x for any base b. Let f (x) = ln x, x > 0. Step 1: Find f (x+h) Step 2: Find f (x + h) f (x) Step 3: Find f ( x h) f ( x) h 3 P a g e

4 Step 4: Find lim h 0 f ( x h) h f ( x). Let s = x/h. Ex: Find derivatives for f (x) = 5 ln x f (x) = x ln x f (x) = 10 ln x f (x) = x 4 ln x 4 = Other Logarithmic and Exponential Functions Logarithmic and exponential functions with bases other than e may also be differentiated. d dx log b x 1 1 ln b x d dx b x b x ln b 4 P a g e

5 Ex: Find derivatives for f (x) = log 5 x f (x) = 2 x 3 x f (x) = log 5 x 4 Note: log 5 x 4 = 4 log 5 x Summary 5 P a g e

6 Application On a national tour of a rock band, the demand for T-shirts is given by p(x) = 10(0.9608) x where x is the number of T-shirts (in thousands) that can be sold during a single concert at a price of $p. 1. Find the production level that produces the maximum revenue. How to find maximum on calculator: 1. y= enter your function 2. graph 3. 2 nd Trace (Calc) 4. Option #4 5. Left bound? Scroll to left hand side of where you think the maximum is. 6. Right bound? Scroll to right hand side of where you think the maximum is. 7. Guess? Place cursor exactly where you think the maximum is. 8. Enter 2. Find the rate of change of price with respect to demand when demand is 25, P a g e

7 Section Derivatives of Products and Quotients Objectives: The student will be able to calculate: the derivative of a product of two functions, and the derivative of a quotient of two functions. Recall: [ ] [ ] Derivatives of Products Theorem 1 (Product Rule) [ ] In words: The derivative of the product of two functions is the first function times the derivative of the second function plus the second function times the derivative of the first function. First times derivative of second plus second times derivative of first Find the derivative 7 P a g e

8 Find the derivative Derivatives of Quotients Theorem 2 (Quotient Rule) [ ] [ ] In words: The derivative of the quotient of two functions is the bottom function times the derivative of the top function minus the top function times the derivative of the bottom function, all over the bottom function squared. Find the derivative 8 P a g e

9 Find the derivative *Not every quotient needs to be differentiated using the quotient rule. Find the derivative Find the derivative Tangent Lines Let f (x) = (2x 9)(x 2 + 6). Find the equation of the line tangent to the graph of f (x) at x = 3. 9 P a g e

10 Higher Order Derivatives: f (x) = the derivative of f(x) = first derivative f (x) = the derivative of f (x) = second derivative f (x) = the derivative of f (x) = third derivative f 4 (x) = the derivative of f (x) = the forth derivative Given f (x) = find: f (x) = f (x)= f 4 (x) = Summary Product Rule: [ ] Quotient Rule: [ ] [ ] 10 P a g e

11 Section The Chain Rule Objectives: The student will be able to form the composition of two functions. The student will be able to apply the general power rule. The student will be able to apply the chain rule. Composite Functions Definition: A function m is a composite of functions f and g if m(x) = f [g(x)] The domain of m is the set of all numbers x such that x is in the domain of g and g(x) is in the domain of f. General Power Rule We have already made extensive use of the power rule: Now we want to generalize this rule so that we can differentiate composite functions of the form [u(x)] n, where u(x) is a differentiable function. Is the power rule still valid if we replace x with a function u(x)? Example: Let u(x) = 2x 2 and f (x) = [u(x)] 3 = 8x 6. Which of the following is f (x)? (a) 3[u(x)] 2 (b) 3[u (x)] 2 (c) 3[u(x)] 2 u (x) 11 P a g e

12 Generalized Power Rule What we have seen is an example of the generalized power rule: If u is a function of x, then d n n 1 du u dx nu dx Chain Rule We have used the generalized power rule to find derivatives of composite functions of the form f (g(x)) where f (u) = u n is a power function. But what if f is not a power function? It is a more general rule, the chain rule, that enables us to compute the derivatives of many composite functions of the form f (g (x)). Chain Rule: If y = f (u) and u = g(x) define the composite function y = f (u) = f [g(x)], then du dy dx dy du dx, provided dy du and du dx exist. Alternatively: The composite function defined by f(g(x)) has a derivative given by. That is differentiate the outside function leaving the inside alone then multiply by the derivative of the inside Find the derivative 12 P a g e

13 Find the derivative Find the derivative 13 P a g e

14 Examples find the derivative of each of the following: 14 P a g e

15 15 P a g e

16 Examples for Product and Chain: 16 P a g e

17 Section Implicit Differentiation Objectives: The student will be able to Use special functional notation, and Carry out implicit differentiation. Function Review and New Notation So far, the equation of a curve has been specified in the form y = x 2 5x or f (x) = x 2 5x (for example). This is called the explicit form. y is given as a function of x. However, graphs can also be specified by equations of the form F(x, y) = 0, such as F(x, y) = x 2 + 4xy 3y This is called the implicit form. You may or may not be able to solve for y. Explicit and Implicit Differentiation Consider the equation y = x 2 5x. To compute the equation of a tangent line, we can use the derivative y = 2x 5. This is called explicit differentiation. We can also rewrite the original equation as F(x, y) = x 2 5x y = 0 and calculate the derivative of y from that. This is called implicit differentiation. 17 P a g e

18 Example: y is expressed as a function of x explicitly y is expressed as a function of x implicitly Find of? Now we can see that Find of? In this case it is very difficult to express y as a function of x explicitly. So we use implicit differentiation remembering that y is a function of x it s just that we can t express that function explicitly: y=f(x) Step1: Replace y with f(x) in the equation Step2: Differentiate both sides 18 P a g e

19 Step3: Simplify notation Step4: Solve the equation for y *Note: implicit differentiation can produce an expression that involves both x and y. Ex: If y=f(x) is a differentiable function of f such that find f (x) 19 P a g e

20 Ex: Find the equation of the tangent line to the graph of at the point (1, -2) Ex: If y=f(x) is a differentiable function of f such that x 2 3xy + 4y = 0 find f (x) 20 P a g e

21 Ex: Find the equation of the tangent line to the graph of x 2 3xy + 4y = 0 at the point (1, -1) Ex: If y=f(x) is a differentiable function of f such that xe x + ln y 3y = 0 find f (x) 21 P a g e

22 Notes: Why are we interested in implicit differentiation? Why don t we just solve for y in terms of x and differentiate directly? The answer is that there are many equations of the form F(x, y) = 0 that are either difficult or impossible to solve for y explicitly in terms of x, so to find y under these conditions, we differentiate implicitly. Also, observe that: Ex: If y=f(x) is a differentiable function of f such that find f (x) 22 P a g e

23 Find the equation of the tangent line to the graph of at the point (1, 1) 23 P a g e

24 Section Related Rates Objectives: The student will be able to solve related rate problems and applications. Introduction Related rate problems involve three variables: an independent variable (often t = time), and two dependent variables. The goal is to find a formula for the rate of change of one of the independent variables in terms of the rate of change of the other one. These problems are solved by using a relationship between the variables, differentiating it, and solving for the term you want. Recall: One application of the derivative is rate of change if y=f(x), then the (instantaneous) rate of change of y with respect to x is. Suppose we have two variables (x, y) that are both function of time, t. x=f(t) and y=g(t) If we have an equation that relates x and y, then we can find an equation that relates their rate of change. In the problems in this section we will usually have: 1. An equation which somehow relates x and y 2. Knowledge of the rate of change of one of the variables with respect to time. 3. We may be asked to find the rate of change of the other variable with respect to time. 4. Implicit differentiation of the equation in step one. 5. We will plug in known values to find the unknown rate of change we seek in step P a g e

25 The problems will usually be expressed as word problems but let s do one without the words. If and when x=-2 and y=1 find Solving Related Rate Problems Step 1. Make a sketch. Step 2. Identify all variables, including those that are given and those to be found. Step 3. Express all rates as derivatives. Step 4. Find an equation connecting variables. Step 5. Differentiate this equation. Step 6. Solve for the derivative that will give the unknown rate. 25 P a g e

26 Ex: All edges of a cube are expanding at a rate of 3cm per second. How fast is the volume changing when each edge is at 1cm and at 10cm? Ex: A fire has started in a dry open field and spreads in the form of a circle. The radius of the circle of fire increases at a rate of 6 ft/min. Find the rate at which the fire area is increasing when the radius is 150ft. 26 P a g e

27 Ex: A weather balloon is rising vertically at the rate of 5 meters per second. An observer is standing on the ground 300 meters from the point where the balloon was released. At what rate is the distance between the observer and the balloon changing when the balloon is 400 meters high? 27 P a g e

28 Ex: A rock is thrown into a still pond and causes a circular ripple. If the radius of the ripple is increasing at 2 feet per second, how fast is the area changing when the radius is 10 feet? [Use A = πr 2 ] 28 P a g e

29 Related Rates in Business Suppose that for a company manufacturing transistor radios, the cost and revenue equations are given by C = 5, x and R = 10x 0.001x 2, where the production output in 1 week is x radios. If production is increasing at the rate of 500 radios per week when production is 2,000 radios, find the rate of increase in (a) Cost (b) Revenue 29 P a g e