Math for Non-Geeks/ Examples for derivatives

oben In this chapter, we want to summarise the most important examples of derivatives. The rules for derivatives will allow us for computing derivatives of composite functions. Table of important derivatives In the following table , and is given. We also define , and . function termterm of the derivative functiondomain of definition of the derivative Examples for computing derivatives Now we will calculate some examples of derivatives from the table above. Often it comes down to determining the differential quotient of the function, i.e. a limit value. But sometimes it is also useful to use the calculation rules from the chapter before. Constant functions We start with some simple derivatives: Derivative of a constant function Every constant function is differentiable on all of with derivative . Let . Then there is Power functions with natural numbers as powers Now we turn to the derivative of power functions with natural powers. First we will deal with a few special cases: Derivative of the identity and the square function The functions and are differentiable on all of . Further there is for : as well as For the derivative of we used the 3rd binomial formula . Derivative of a power function Compute the derivative of For there is Instead of using the identity , we could also have calculated using polynomial division. Now we turn to the general case, i.e. the derivative of for : Derivative of power functions The power function is for differentiable on all of . For all there is For , so there is We used the geometric sum formula and the continuity of the polynomial function . Polynomials and rational functions Using the calculation rules for derivatives we can now calculate the derivatives of polynomial functions and rational functions: Derivative of polynomial functions Let with and be a polynomial function of degree . Then is differentiable on all of , and for there is Using the derivative rule for the multiple of a function , every single summand of the polynomial is differentiable on . With the summation rule we can derive every polynomial function term by term on and obtain for : with the derivative of the zeroth summand disappearing. In particular, it follows for and that linear and quadratic functions are differentiable on all of . Derivative of rational functions Let with and a rational function defined on . Show that is differentiable on , and calculate the derivative. Numerator and denominator of are polynomials. Since the denominator is non-zero on and polynomials are differentiable, it follows from the quotient rule that is differentiable on . Further there is for : Power functions with integer powers We can already differentiate power functions with natural powers. Now we investigate those with negative integer exponents. Derivative of the hyperbolic function The power function is differentiable on and there is for . Derivative of Prove that the power function is differentiable on and compute its derivative. For there is In the general case with there is Derivative of the power function with negative integer powers The power function is differentiable on , and for there is For there is Derivative of the power function Prove using the quotient rule For there is by the quotient rule Remark: Of course we can also apply the inverse rule directly, and thus get the same result Let us look again at the derivatives rule in the last case, i.e. for . If we put , we get . The derivative rule is hence the same as for with . So we can summarize the two cases and get Derivative of the power function with natural powers For the power function is differentiable on . For there is then In the case of it is even differentiable on all of . Root functions Now we investigate the derivative of root functions. We start again with the simplest case: Derivative of the square root function The square root function is differentiable on and for there is Question Why is the square root function in not differentiable, although it is defined and continuous there? For the differential quotient there is So it does not exist. Hence, we have non-differentiability. Derivative of the cubic root function Compute the derivative of the cubic root function For there is Now let us consider the general case of the -th root function. Here there is Derivative of the -th root function Let . Then the -th root function is differentiable on , and for there is For there is This can now be generalized Derivative of the generalized root function For and , the generalized root function is differentiable on , and for there is Since on die functions and are differentiable, the chain rule implies at that For and and the power function with rational exponent was defined as So for we also have the derivative rule The (generalized) exponential function and generalized power functions In this section we prove that the derivative of the exponential function is again the exponential function. So we can determine the derivative of the generalized exponential and power function. Derivative of the exponential function The exponential function is differentiable on , and for there is For this derivative it is more useful to use the method Because in this case we know the limit value Furthermore, we need the functional equation of the exponential function For there is Using the chain rule, the derivatives of the generalized exponential function for and the generalized power function for can be calculated: Derivative of the generalized exponential function For the generalized exponential function is differentiable on , and for there is For there is Derivative of the generalized exponential function For the generalized exponential function is differentiable on , and for there is Derivative of the generalized exponential function Prove that the derivative of the generalized power function at is . For the chain rule yields Logarithmic functions Now we turn to the derivative of the natural and generalized logarithm function. Since the natural logarithm is the inverse of the exponential function, we can deduce its derivative directly from rule for derivatives of the inverse function: Derivative of the natural logarithm function The natural logarithm function is differentiable on . For there is For the exponential function there is: . So the function is differentiable, and because of strictly monotonically increasing. Furthermore, is surjective. The inverse function is the (natural) logarithm function From the theorem about the derivative of the inverse function we now have for every : The derivative can also be calculated directly using the differential quotient. If you want to try this, we recommend the corresponding exercise. Using the derivative of the natural logarithm function we can now immediately conclude Derivative of the generalized logarithm function For the generalized logarithm function is differentiable on . For there is From the derivative rule for the multiple of a function, we get that for all : If the derivative of the natural logarithm is not available, we can calculate it using the theorem of the derivative of the inverse function. Trigonometric functions Sine Derivative of the sine function The sine function is differentiable. For all there is: For there is Cosine Derivative of the cosine function The cosine function is differentiable with Tangent Derivative of the tangent function The tangent function is differentiable on , and for there is Since for , the function is differentiable by the quotient rule, and for there is Derivative of the cotangent function The cotangent function is differentiable on , and for there is Since for , the function is differentiable by the quotient rule, and for there is Alternative solution: The derivatives of secant and cosecant can be found in the corresponding exercise. arc-functions Using the rule for derivatives of the inverse function we can differentiate the arc-functions (which are inverses of sine, cosine, etc.) arcsin and arccos Derivative of the arcsin/arccos function The inverse functions of the trigonometric functions , are differentiable with Note: and are defined and continuous on , but only differentiable on . Derivative of : For the sine function there is: . So the function is differentiable, and since for all , it is strictly monotonically increasing on this interval. Further, . So is surjective. The inverse function is the arc sine function From the theorem about the derivative of the inverse we now have for every : Derivative of : For the cosine function there is: . So the function is differentiable, and because of , strictly monotonically decreasing. Further, . So is surjective. The inverse function is differentiable according to the theorem about the derivative of the inverse function, and for every there is: arctan and arccot Derivative of the arctan/ arccot function The inverse functions of the trigonometric functions , are differentiable, and there is For the tangent function there is: . So the function is differentiable and strictly monotonically increasing. Further, . So is surjective. The inverse function is hence differentiable, and now for there is: Hyperbolic functions And finally, we determine the derivatives of the hyperbolic functions , and : Derivative of hyperbolic functions The functions are differentiable, and there is The derivatives follow directly from the calculation rules. We show only the derivative of . The other two are left to you for practice. According to the factor and difference rule for all is differentiable, and there is Derivative of and Prove that and are differentiable with and Derivative of : According to the factor and sum rule, is differentiable for all , and there is Derivative of : is differentiable on all of by the quotient rule and there is unten