Question

Find all the second partial derivatives in each of the following cases: (a) \(z=x y\) (b) \(z=7 x y\) (c) \(z=8 x+9 y+10\) (d) \(z=8 y^{2} x+11\) (e) \(z=-2 y^{3} x^{2}\) (f) \(z=x+y\)

Short answer

Question: Find the second partial derivatives of the following functions: (a) \(z = xy\) (b) \(z = 7xy\) (c) \(z = 8x + 9y + 10\) (d) \(z = 8y^2x + 11\) (e) \(z = -2y^3x^2\) (f) \(z = x + y\) Answer: (a) z_xx = 0, z_yy = 0, z_xy = 1, z_yx = 1 (b) z_xx = 0, z_yy = 0, z_xy = 7, z_yx = 7 (c) z_xx = 0, z_yy = 0, z_xy = 0, z_yx = 0 (d) z_xx = 0, z_yy = 16x, z_xy = 16y, z_yx = 16y (e) z_xx = -4y^3, z_yy = -12y^2, z_xy = -12y^2x, z_yx = -12y^2x (f) z_xx = 0, z_yy = 0, z_xy = 0, z_yx = 0

Step-by-step solution

(a) Find the second partial derivatives of \(z = xy\)

First derivatives: z_x = y z_y = x Second derivatives: z_xx = 0 z_yy = 0 z_xy = 1 z_yx = 1

(b) Find the second partial derivatives of \(z = 7xy\)

First derivatives: z_x = 7y z_y = 7x Second derivatives: z_xx = 0 z_yy = 0 z_xy = 7 z_yx = 7

(c) Find the second partial derivatives of \(z = 8x + 9y + 10\)

First derivatives: z_x = 8 z_y = 9 Second derivatives: z_xx = 0 z_yy = 0 z_xy = 0 z_yx = 0

(d) Find the second partial derivatives of \(z = 8y^2x + 11\)

First derivatives: z_x = 8y^2 z_y = 16yx Second derivatives: z_xx = 0 z_yy = 16x z_xy = 16y z_yx = 16y

(e) Find the second partial derivatives of \(z = -2y^3x^2\)

First derivatives: z_x = -4y^3x z_y = -6y^2x^2 Second derivatives: z_xx = -4y^3 z_yy = -12y^2 z_xy = -12y^2x z_yx = -12y^2x

(f) Find the second partial derivatives of \(z = x + y\)

First derivatives: z_x = 1 z_y = 1 Second derivatives: z_xx = 0 z_yy = 0 z_xy = 0 z_yx = 0

Key concepts

Partial Differentiation

Partial differentiation is a fundamental technique in multivariable calculus where we find the derivative of a function with respect to one variable while keeping other variables constant. Consider a function with two variables, like the case (a) in our exercise where we have the function z = xy. To find the partial derivative with respect to x, we differentiate z with respect to x, treating y as a constant. So for the first derivative, we get z_x = y. Similarly, for the partial derivative with respect to y, we treat x as a constant and we get z_y = x.

The concept extends to finding second partial derivatives, which is like taking the derivative of the derivative. From the exercise, second partial derivatives of z = xy with respect to both x and y are zero since the first derivatives are constants. This process highlights curvature and inflection points in varying directions of a surface or function.

Multivariable Calculus

Multivariable calculus extends the concept of single-variable calculus to functions of multiple variables. The idea is to study how such functions change—essentially uncovering the terrain of multi-dimensional landscapes. The second partial derivatives are particularly important as they contribute to the understanding of the function's concavity in different directions, and the mixed derivatives (z_xy and z_yx) assess how the function curves when moving diagonally across the surface.

In our exercise, such as case (b) where z = 7xy, the mixed second partial derivatives z_xy = 7 and z_yx = 7 tell us that the surface's curvature is the same in both mixed directions. This symmetry in mixed second partial derivatives is a reflection of Clairaut's theorem—a crucial aspect of multivariable calculus.

Mathematical Analysis

Mathematical analysis involves the study of limits, derivatives, and integrals of functions. Second partial derivatives play a critical role in what's known as the second derivative test for local extrema on functions of several variables. In this test, if we consider the Hessian matrix—a square matrix of all the second partial derivatives—we can analyze the definiteness of this matrix to determine if a given point is a local minimum, local maximum, or a saddle point.

Looking at exercises like (e) z = -2y^3x^2, second partial derivatives such as z_xx = -4y^3 and z_yy = -12y^2 would contribute to the Hessian matrix. By determining the signs and definiteness of the matrix, we can deduce the nature of the point in question with respect to the original function's landscape.

Engineering Mathematics

In engineering mathematics, second partial derivatives are essential for optimizing designs, analyzing systems, and solving physical problems. They contribute to the description of wave behavior, heat conduction, electromagnetic fields, and much more. For example, in the function from case (d) z = 8y^2x + 11, examining the second partial derivatives, z_yy = 16x, isn't just an academic exercise but can represent the rate at which a physical quantity changes directionally in space.

Understanding these concepts is pivotal for engineers as they analyze multidimensional systems. For instance, the partial derivative z_xy or z_yx would be critical in assessing how two different forces or influences interact with each other in a dynamic system. Second partial derivatives provide deeper insights into the behavior and interactions of complex engineering systems.