M AT 265 Study Guide for TEST 2 Section 2.5: The Chain Rule• Know how to apply the chain rule. Remember that we always work from the outside to the inside: To differentiate the function Fx=f gxwe differentiate the outer function f [at the inner function g(x)] and then we multiply by the derivative of the inner function: ddxfg x=f'g x⋅g ' x (see Examples 1-6, hw 1-10, 12 )• Be especially careful in those cases where you have to use the chain rule twice (or more) (see Examples 7 and 8, hw 11, 15, 16) or when you have to use the chain rule together with the product and/or quotient rule (hw 9, 13, 14)• Practice on finding higher derivatives (hw 18-20) and finding the equation of the tangent line at a given value of x (hw 17, 22).• Know how to apply the chain rule (as well as the other derivative rules) for functions given by tables (hw 21). Section 2.6: Implicit Differentiation • Know how differentiate implicitely. Remember that , since you are assuming y is a function of x, you will most likely have to use the chain rule and/or the product or quotient rules. (see Examples 1-3, hw 1-10).• Know how to find the second derivative implicitely (see Example 4, hw 11).Section 2.7: Related Rates• In a related rates problem the idea is to compute the rate of change of one quantity in terms of the rate of change of another quantity. The procedure is to find an equation that relates the two quantities and then use the Chain Rule to differentiate both sides with respect to time. Always start by identifying the given information and the unknown information (both of these will be derivatives). To write a relation between the two quantities you might need to use geometry. Read all the examples in this Section and review all the homework problems.Section 2.8: Linear Approximations and Differentials• Linear Approximation: Know the formula for the linear approximation (or linearization) of a function f at a : Lx=faf'ax−a(hw 1, 2). Understand that this is just the equation of the tangent line and can be use to approximate values of f for x near a. (see Examples 1 and 2, hw 3-5, 10)• Differentials: • If y = f(x), then the differential dy is defined by dy=f' x dx(hw 7). Understand the geometrical meaning of the differential: dy represents the amount that the tangent line rises or falls when x changes by the amount dx. Thus the differential is an approximation to the change in y which we denote by ∆y (hw 6).• Understand how differentials can be used to estimate the errors that occur because of approximate measurements (hw 8,9)Section 3.1: Exponential Functions• Know how the graph of the exponential function y=axlooks like for 0 < a < 1 and for a > 1 and be able to determine the limits for x approaching ∞and/or −∞ .(see Example 1 and 2, hw 1-8)Section 3.2: Inverse Functions and Logarithms• Know how to determine if a function is one to one (Examples 1 and 2) and how to find the inverse of a one to one function (Example 4, hw 1)• Understand that the logarithmic function is the inverse of the exponential function. Review the Laws of Logarithms (Example 7, 11, hw 2-7)• Know how the graph of a logarithmic function looks like and be able to determine limits at infinity and vertical asymptotes (Example 8, 13, hw 10, 11).• Know how to solve exponential and logarithmic equations (Example 9 and 10, hw 10, 11)Section 3.3: Derivatives of Logarithmic and Exponential Functions• Know the formula for the derivative of the logarithmic function: ddxlogax=1xlnaand the particular case: ddxlnx=1xand how to use these formula together with the chain rule and product/quotient rule (see Examples 1-6, hw 1-16)• Understand how to use Logarithmic differentiation to simplify the calculation of the derivative of complicated functions: start by taking ln of both sides, apply the Laws of Logarithm to get rid of products and quotients and then differentiate implicitely. Finally, solve the resulting equation for y' (Example 7, hw 17-21 ).• Know how to use logarithmic differentiation to differentiate functions of the form y=fxgx(see Example 11, hw 19-21)• Know the formula for the derivative of the exponential function: ddxax=axlnaand the particular case: ddxex=ex.(Examples 8, 9, 10, hw 22-32, 35,36)• Practice on doing implicit differentiation for expressions involving logarithms and exponentials (hw 33, 34)Section 3.5: Inverse Trigonometric Functions• Know the definition, the domain and the range for all inverse trigonometric functions. (Examples 1, 2,3, hw 1,2)• Know the limits at infinity for arctan x and related functions (Example 4).• Know the formulas for the derivatives and be able to derive them using implicit differentiation and known trig identities. ddxarcsinx=11−x2 ddxarctanx=11x2 ddxarcsecx=1xx2−1ddxarccosx=−11−x2 ddxarccotx=−11x2 ddxarccscx=−1xx2−1 • Practice using the formula for the derivaties of inverse trig functions together with the chain rule and/or product and quotient rules. (Example 5, hw 3-10) Review ALL the homework and notes. Read all the Examples in the textbook listed above. For additional practice login asGuest Login in WeBWorK. Click HERE for extra practice on derivatives with ANSWERS ANSWERS (problems 1-42, 49-51, 55-60). You can also work out the assigned homework problems from the