Math for Non-Geeks/ Exercises: Derivatives 1

oben Computing derivatives with differential quotients Exercises: derivative and differentiability Differentiable power function Show that the power function is differentiable at and compute the derivative. What is the derivative of at any ? The differential quotient of at is given by So is differentiable at , with derivative . For a general there is Derivative of a product function Let defined by Determine . There is For we have used that is continuous as a product of the continuous functions for . Derivative of a function with case distinction Check whether the following functions are differentiable at . Part 1: Since oscillates very quickly between and , just like for , it is to be expected that at is not continuous. For this purpose we consider the null sequences and . For these sequences and So does not exist. According to the sequence criterion is therefore not continuous at zero and thus not differentiable. Part 2: The function is continuous at zero by the sequence criterion, as . So we can consider the differential quotient: In Part 1 we have shown that this limit value does not exist. Therefore is also not differentiable at zero. Criterion for non-differentiability of a general function at zero Let . Show that: If for some and , then is not differentiable at zero. There is since So does not exist. Determining limits with the differential quotient Let be differentiable at . Show that the following limits exist for Since in is differentiable, there is and In addition, from the exercises within the article "derivatives", we know that The idea is to transform the limits so that we can calculate them using the differential quotient. Part 1: Because there is also . So Part 2: With and there is also and . Hence, Part 3: Here we need the "original" differential quotient : Implication of differentiability Let be differentiable at . Further, let and be sequences with for all , as well as . Show that then, there is Additional question: Does the converse statement also hold? I.e. does the limit value with sequences and as above exist, so is differentiable at , and is equal to this limit? Hint: Show first that There is Since now the product of a bounded sequence and a null sequence converges to zero, there is with the calculation rules for sequences Concerning the additional question: The converse is false. Let us consider the following function, which is not continuous at (and therefore not differentiable): Then, there is for all null sequences and with : Exercises: examples for derivatives Derivatives of linear and quadratic functions Determine (using the definition) the derivative of a linear function and of a quadratic function with . 1. linear function: For there is 2. quadratic function: For there is Derivative of the logarithm function Compute the derivative of the natural logarithm function directly, using the differential quotient. 1st way: For there is Now for we have the inequality If we swap the roles of and , then there is Since the left and right-hand sides of the inequality for converge towards , the squeeze theorem implies 2nd way: -method For there is Computing the derivatives of hyperbolic functions and Determine die derivatives of the following functions using the differential quotient Part 1: Let . Then, there is Alternative proof: Part 2: Let . Then, there is Alternative proof: Part 3: Let . Then, there is So Alternative proof: Computation rules for derivatives Applying the computation rules derivatives of a power function Show by induction in , that the power function is differentiable with Induction base If , then there is Induction assumption For with , there is Induction step Let . Then is differentiable by induction assumption and the product rule. For there is Derivatives of secant and cosecant The functions (secant) and (cosecant) are defined as follows: as well as Determine their domain of definition and all derivatives. Part 1: secant domain of definition: is well-defined Derivative: Since is differentiable on , there is with the chain rule Part 2: cosecant domain of definition: is well-defined Derivative: Since is differentiable on , there is with the chain rule Computing derivatives Determine the domain of definition of the following functions, as well as their derivatives Part 1: domain of definition: Derivative: For there is with the product rule Part 2: domain of definition: , as Derivative: For there is with the chain rule Part 3: domain of definition: Derivative: For there is with the chain- and product rule Part 4: domain of definition: , since there must be Derivative: For there is with the quotient rule Part 5: domain of definition: Derivative: For there is For there is Further there is as well as So we have Concluding all three cases, we get for derivatives of exponential functions Determine the derivatives of the following functions on their domain of definition () For the function we may leave out the bracket, since in general is well-defined. Part 1: There is . The function is differentiable with . Hence is differentiable by the chain- and product rule and for there is Part 2: There is . Hence is differentiable by the chain- and product rule and for there is Part 3: There is . Hence is differentiable by the chain- and product rule and for there is Part 4: There is . Hence is differentiable by the chain- and product rule and for there is Part 5: There is . The function is differentiable with . Hence is differentiable by the chain- and product rule and for there is Proof of sum formulas using the derivative Proofs by means of binomial theorem (missing) that for all : Use the binomial theorem and set . Then take the derivative on both sides. For the binomial theorem reads for and . Now the left-hand side of the equation is a polynomial and the right-hand side is a power function . Both sides are therefore differentiable on with and Since there is also . So in particular and Logarithmic derivatives Determine the logarithmic derivatives of the following functions with Proof of computational laws Alternative proof of the product rule Prove that for differentiable the product rule holds - by using the chain rule. Hint: There is: The function is differentiable on with By der chain rule, we hence have that is differentiable with for all . Using the hint, we get with the factor- and sum rule Special case of the chain rule Derive a general derivative formula for the following function: If are differentiable. There is with and for all . The function is differentiable by the product rule with By the chain rule, also is differentiable, and there is Computational laws for logarithmic derivatives For two differentiable functions and without zeros there is for and for and unten