Higher-Order Derivatives
Higher-order derivatives refer to the derivatives of a function beyond the first derivative. They represent the rate of change of the rate of change, and so on, of the original function. For example, the second derivative represents the curvature of the function, while the third derivative represents the rate of change of curvature, and so forth.
4 Key excerpts on "Higher-Order Derivatives"
eBook - ePub- Adil H. Mouhammed(Author)
- 2020(Publication Date)
- Routledge(Publisher)
...Derivatives Similar to a higher-order differentiation of functions of one variable, the multivariate functions can be differentiated to higher orders such as first, second, third, and so on as long as the given function permits that. Literally, for example, the second-order partial differentiation measures the rate of change of the rate of change of the dependent variable with respect to a change in an independent variable, holding all other independent variables constant. Suppose we have the function Y = f(X 1, X 2, X 3,..., X n) the following notations are used to indicate the second-order partial derivatives: f x1x1, f x1x2, f x1x3, f x1xn, f x2x2, f x3x3, and f xnxn. One should note that the notations f x1x1, ξ 2x2, Ç 3x3, and z nxn indicate the direct second order partial derivatives, whereas f x1x2, f x1x3 „ f x1xn, f x2xn and so on indicate the cross-partial Higher-Order Derivatives. In addition, and according to Young’s theorem, the second-order cross-partial derivatives such as f x1x2 and f x2x are equal. Similarly, f x3x1 and f x1x3 are equal as well. Example 1: Find all second-order partial derivatives for Y = 4x 1 x 2. Solution: The first partial derivatives are f x 1 = 4 x 2 and f x 2 = 4 x 1. the second direct partial derivatives are f x 1 x 1 = 0 and f x 2 x2 = 0, and the second cross-partial derivatives are f x 1 x 2 = 4 and f x 2 x 1 = 4. Example 2: Find all second-order partial derivatives for Y = 5x 1 2 x 2 2 + x 1 +. 3x 2. Solution: f x 1 = 1 0 x 1 x 2 2 + 1 and f x 2 = 10 x 1 2 x 2 + 6 x 2. f x 1 x 1 = 10 x 2 2 and f x 2 x 2 = 10 x 1 2 + 6. And f x1x2 = 20x 1 x 2 and f x2x1 = 20x 1 x 2. Example 3: Find all partial derivatives for Z = e xy. Solution: Z x = ye xy and...
- F. Xavier Malcata(Author)
- 2020(Publication Date)
- Wiley(Publisher)
...A concave curve (i.e. exhibiting a concavity facing upward) will accordingly hold a positive d 2 f / dx 2, whereas a larger value of d 2 f / dx 2 entails a more pronounced curvature (or deviation from a straight line) – and vice versa. Higher‐order derivatives can similarly be defined, according to (10.52) that essentially mimics Eq. (10.49), again at the expense of derivatives with immediately lower order; one consequently obtains (10.53) as per the definition, which degenerates to (10.54) After insertion of expressions describing the previous derivative and iteration of the process until getting to f { x } explicitly, one eventually obtains (10.55) – or, using a more condensed notation resorting to binomial coefficients, (10.56) the practical usefulness of derivatives with order n ≥ 3 is, nevertheless, marginal – because they become excessively sensitive to disturbances in the departing function f { x }, oftentimes obtained as fit to experimental data subjected themselves to random error. 10.2.1.2 Partial Derivatives If a multivariate function is at stake, a rationale similar to regular derivatives applies – yet more than one derivative can now be calculated; in the specific case of a bivariate function f { x,y }, one accordingly gets (10.57) and (10.58) for the two partial (first‐order) derivatives with regard to x and y, respectively. The former may be thought as the ordinary derivative of f { x,y } with respect to x, obtained by treating y as a constant; likewise, the partial derivative of f { x,y } with respect to y may be found by treating x as a constant, and then calculating the ordinary derivative of f { x,y } with respect to y. In both cases, the variable that is supposed to be held constant in the differentiation is indicated as subscript – a notation particularly useful when more than two independent variables are under scrutiny. By the same token, one may define higher order derivatives – namely, (10.59) based on Eq...
eBook - ePub- Arsen Melkumian(Author)
- 2012(Publication Date)
- Routledge(Publisher)
...Sometimes we may be interested in finding the third, fourth or n th derivative of a function. If y = f (x) is differentiable n times, then we let f (n) (x) denote the n th derivative of y = f (x) at a given point x. E XAMPLE 4.5 Let w (x) = x 7 + 2 x 5 + 19. Find the first four derivatives of w (x). Solution : Using Der1 – Der8, we have Note that w (x) = x 7 + 2 x 5 + 19 is a polynomial function and as such can be differentiated infinitely many times. However, all derivatives of order eight or above are equal to zero. In general, if f (x) is a polynomial function of degree n, then the first n derivatives of f (x) are non-zero and all derivatives of higher order vanish. E XAMPLE 4.6 Consider a firm producing output with just one input, labor L. The firm’s total product Q is a function of its only input: Q (L) = AL 1− b. Find Q ′(L) and Q ″(L). Solution : Using Der1 – Der8, we have Economists refer to the expression as the marginal product of labor (MPL). The MPL is defined as the change in total product from expanding labor input by one unit while holding everything else constant. By assumption MPL is always positive. Functions that are continuous and have continuous first-order derivative belong to a class of functions called C 1. This notation is commonly used in mathematics and economics books. The index 1 in the notation C 1 indicates that all the derivatives up to the first order exist and are continuous. Similarly, notation C 2 stands for functions that are twice differentiable and their second derivatives are continuous. 4.2.1 Partial derivatives Often economists work with multivariate functions such as We can differentiate the function above with respect to each of its three independent variables as follows: Note that whenever we differentiate a multivariate function with respect to one independent variable, we consider the remaining variables to be constant...
eBook - ePub- Gregory Hill, Mel Friedman(Authors)
- 2013(Publication Date)
- Research & Education Association(Publisher)
...In symbols, this means and substituting e x for y gives The graph of f (x) = e x is shown below with the line tangent to the function at the point (0, 1). Notice that the slope of the line appears to be 1. This will be accepted without formal proof. Figure 4.1 Recall the alternate definition of a derivative, Let a = 0, replace x with h, and apply it to the function f (x) = e x. Then the slope of the tangent to the graph at x = 0 is Now examine the derivative of e x by definition. Three more key derivatives are given here without proof. If a positive constant and a ≠ 1, then If y = ln(x), then If a is a positive constant and a ≠ 1, then The last derivative given does not really need to be memorized. It can be easily established by using the change of base rule for logarithms, Since ln(a) is a constant, taking the derivative of is no different than taking the derivative of ln(x) and multiplying the result by As with the trigonometric functions, the properties of exponential and logarithmic functions will be crucial to recall, especially in multiple-choice situations where a simplified choice of the derivative may be offered, as in Example 4.11. EXAMPLE 4.11 If f (x) = log 3 (x), then f ′(2) = (A) log 3 2 (B) log 3 1 (C) (D) (E) SOLUTION It seems that the correct solution is not offered, but (D) is the proper choice. Because of the property of logarithms, This example shows the importance of knowing the properties of logarithms and exponential functions. 4.6 Higher-Order Derivatives If the derivative of a function exists and is itself differentiable, it is possible to take the derivative of the derivative. Naturally, this is called the second derivative. If the first derivative represents the instantaneous rate of change of the function at a point, the second derivative represents the rate of change of the rate of change. The significance of this idea will be explored in the next chapter...
