98 CHAPTER 3. DIFFERENTIAL CALCULUS3.4.10 Logarithmic differentiationTextbook pages 231-233In the previous examples, we saw that taking the derivative of the logarithm of a com-plicated function involving products and powers is actually very easy. By comparison, takingthe deriva tive of the function itself could be more complicated. Based upo n this idea, we nowintroduce the concept of logarithmic differfentiation. The following example will illustratehow powerful this method is.Example: Find the derivative off(x) =e2x(2x − 1)6(x + 5)3(4 − 7x)Idea:• Take the logarithm of both sides of this equality, and expand the log.• Take the derivative of both sides of this equality• Multiply both sides by f (x) to obtain f′(x)3.4 DERIVATIVE RULES 99Example: Find the derivative off(x) =2x(x + 1)2(x2+ 2x − 1)3(2 + 7x)Example: The logarithmic differentiation is also useful to find derivatives of complicatedfunctions such as f(x) = xx.Proof of the power rule in the general case: Earlier in the lecture, we proved thepower rulefor positive integer vales of α. We are now able to prove it in the general case, using loga-rithmic differentiation.100 CHAPTER 3. DIFFERENTIAL CALCULUS3.4.11 Implicit differentiationTextbook pages 200-203So far, we have only cons idered taking derivatives of functions defined explicitly as y = f (x).In that case, y is a genuine, uniquely defined function of x, and the derivative f′(x) is theslope of the tangent to the graph y = f(x) at every point.However, in many cases one may also like to know the tangent to curves which are notdefined as functions of x. For example, in the cas e of a circle ,In this cas e , x and y are related by the above equation, or in other words, y is defined im-plicitly in terms of x through this equation.Question: What is the tangent to the circle x2+ y2= 1 at the point (√2,√2)?How can we find the equation for the tangent in this ca se? We can consider two meth-ods: the direct method, which is more complicated (and sometimes impossible) , and themethod o f implicit differentiation, which is easier (and always possible).Direct Method: If y is not defined in terms of x, the first thing to do is to solve fory. In the case of the circle,Then ta ke the derivative of y(x):Once we have the derivative, we have the slope of the tangent, so we can calculate the lineequation with the point-slope formula:3.4 DERIVATIVE RULES 101Problem: If we had a mo re complicated e quation relating x and y, it would be impossibleto solve for y, so what could we do in this case?Idea: Although technically we cannot write y = f(x) (because the cir c le is not the graph o fa function), we will still do it and then take the derivative of the whole equation:• Step 1: Take the equation and substitute f (x) for y.• Step 2: Take the derivative of the equation• Step 3: Substitute back y for f (x)• Step 4: If f′(x) is needed, solve for it.This gives the slope of the curve at any point (x, y). Then we use the point given (√2,√2)to get the slope at the point desired.Interest of the method: This method can be used for any complicated curve! A famouscurve is called the Bifolium, and is given by the equation(x2+ y2)2= 4x2y102 CHAPTER 3. DIFFERENTIAL CALCULUSQuestion 1: Check that the point Aq34+1√2,12is on the Bifolium.Question 2: What is the slope of the tang e nt to the Bifolium at any point (x, y)?Question 3: What is the equation of the tangent to the Bifolium at the point A?3.4 DERIVATIVE RULES 103Question 4: At which points does the Bifolium have a horizontal tangent?104 CHAPTER 3. DIFFERENTIAL CALCULUSCheck your understanding of Lecture 13• Logarithmic differentiation:– Let f(x) = g(x)α. Using the chain rule, prove that f′(x) = αg′(x)g(x)α−1. Provethe s ame result using logarithmic differentiation (i.e. say ln(f(x)) = ln(g(x)α)and take the derivative on both sides).– Textbook pr oblems page 234 number 63 , 65, 67, 71, 73, 75.• Tangents to the circleConsider the equation x2+ y2= a2. What geometrical object does this correspond to(give a precise description)? Find the slope of the tang ent to this curve at any point onthe curve. If the point with x−coordinate c is on the curve, what is its y-coordinate?Deduce the equation for the tangent to this curve at any point on it.• Tangents to an ellipseConsider the ellipse x2+2y2= 4. What is the y-coordinate of a point with x−coordinate1? Find the equation of the tangent to this ellipse at this point.• Using the implicit differentiation method, find the slope of the tangents to each of thesecurves at any point (x, y) on the curve:– x4+ y4= 2– x2y + 3ey= ln(x)– sin(x) = cos(y)– 3xy + 4y6sin(x) = y sin(y)1053.5 Higher order d erivatives3.5.1 Higher order derivativesHigher o rder derivatives are simply successive derivatives of the same function.Second order derivative:Third order derivative:n-th order derivative:Examples:• f (x) = (2x + 1)3– f′(x) =– f′′(x) =– f(3)(x) =– f(4)(x) =– f(n)(x) =• f (x) = ex– f′(x) =– f′′(x) =106 CHAPTER 3. DIFFERENTIAL CALCULUS– f(3)(x) =– f(n)(x) =3.5.2 Graphical interpretation of the second-order derivativeThe second derivative is the derivative of the derivative. Therefore••Example 1: f (x) = x23.5. HIGHER ORDER DERIVATIVES 107Example 2: f (x) = x3So we note that:Definition:•••3.5.3 Graphing with derivatives (part 3)We now see that to get as much information as possible about a function, we should study• its sign, to know whether it is positive or negative108 CHAPTER 3. DIFFERENTIAL CALCULUS• the sign of its derivative, to know whether it is increasing or decreasing• the sign of its second derivative, to know whether it has a positive or a negative curva-ture.Example 1: f (x) = x(x − 1)(x + 1)3.5. HIGHER ORDER DERIVATIVES 109Example 2: f (x) =x2x2+k2.110 CHAPTER 3. DIFFERENTIAL CALCULUSCheck your understanding of Lecture 14• Second order derivativesCalculate the seco nd order derivative of the following functions:– sin(x)– eaxwhere a is an arbitrary constant– e−x2– x3− 2x + 1• Higher order derivativesCalculate the sucessive derivatives (first, second, third, etc..) of– sin(kx) (where k is an arbitrary constant)– eax(where a is an arbitrary constant)Can you deduce the general formula for the n-th derivative of these functions?• Graphing with derivatives (1)For each of the following, sketch the function f(x) using what you know of bas ic func-tions. Then, by