Math for Non-Geeks/ Exercises: Derivatives 2

oben Derivative of the inverse function Derivative of the inverse function Consider the function Justify that the following derivatives and limits exist and calculate them: and and Part 1: Existence and computation of is differentiable on as a sum of the differentiable functions and with derivative So . Existence and computation of is differentiable on as a sum of the differentiable functions and with derivative Further, , since for . So is strictly monotonically increasing by the monotonicity criterion and hence injective. Further, is bijective. And we have , so , and there is . Hence is differentiable at with Part 2: Computation of There is So Computation of There is . According to the intermediate value theorem, is bijective. For all we thus have So Derivative of a general logarithm function Derivative logarithm function Derivative of a general logarithm function Show that the function with base is bijective and differentiable on . Show further that the inverse function is differentiable on all of with derivative . is bijective and differentiable is continuous on as a composition of continuous functions. Further , since , and there is: as well as By the intermediate value theorem, and is therefore surjective. Moreover, is differentiable according to the chain rule as a composition of differentiable functions, and for all . By the monotonicity criterion, is strictly monotonically increasing, and hence injective. So we have shown the bijectivity of . is continuous on as a composition of continuous functions. Further , since , and there is: as well as By the intermediate value theorem, and is therefore surjective. Moreover, is differentiable according to the chain rule as a composition of differentiable functions, and for all . By the monotonicity criterion, is strictly monotonically decreasing, and hence injective. So we have shown the bijectivity of . exists and is differentiable is bijective on and hence invertible. Further, there is So Since in addition for all , the theorem about inverse function derivatives yields that is differentiable on with derivatives on and area-functions Show that the functions are differentiable, and determine their derivative. Differentiability of : For the cotangent function there is:. Thus the function is differentiable and strictly monotonically decreasing, and thus injective. Further, . Thus is bijective. The inverse function is differentiable according to the theorem on the derivative of the inverse function, and for there is: Differentiability of : The hyperbolic sine function is differentiable with . Thus, it is strictly monotonically increasing, and hence injective. Further, . So it is also surjective. The inverse function is differentiable by the theorem on the derivative of the inverse function , and for there is: Differentiability of : The hyperbolic cosine function is differentiable with on . Thus, it is strictly monotonically increasing, and hence injective. Further, . So it is also surjective. The inverse function is differentiable by to the theorem on the derivative of the inverse function, and for there is: Differentiability of : For the cotangent function there is: . Thus, the function is strictly monotonically increasing, and thus injective. Further, . Thus is bijective. The inverse function is differentiable by the theorem on the derivative of the inverse function, and for there is: Non-differentiable functions at zero Let . Show that: Let with and for all . Then and are not simultaneously differentiable at zero. Let , and let be differentiable at zero. Further let and for all . Then is not differentiable at zero. Part 1: Assuming and are differentiable at zero. Then according to the product rule, is also differentiable at zero, and from we get for all that: ? So and cannot both be differentiable at zero. Part 2: Assume that is differentiable at zero. Then according to the chain rule is also differentiable at zero, and from we get for all that: ? So cannot be differentiable at zero. Derivatives of higher order Exercise 1 Arbitrarily often differentiable function Show that the function is arbitrarily often differentiable and for all there is: The proof goes by induction over : Exercise 2 Exactly one/two/three times differentiable functions Provide an example of a function function that is differentiable, but not continuously differentiable on function function or or or or or or or We can successively proceed with the construction of functions and obtain some for all , with or or Exercise 3 Application of the Leibniz rule Determine the following derivatives using the Leibniz rule for The functions and are arbitrarily often differentiable on . Hence, there is The functions and are arbitrarily often differentiable on . Hence, there is unten