Problem 30 - Calculus Early Transcendentals

Question

Find the four second partial derivatives of the following functions. $$f(x, y)=\cos x y$$

Step-by-Step Solution

Answer

Question: Determine the four second partial derivatives of the function $$f(x, y) = \cos{xy}$$. Answer: The four second partial derivatives are: 1. $$f_{xx} = -y^2\cos{xy}$$ 2. $$f_{yy} = -x^2\cos{xy}$$ 3. $$f_{xy} = -x\cos{xy}$$ 4. $$f_{yx} = -y\cos{xy}$$

Step 1: Find partial derivative with respect to x (f_x)

Find the partial derivative of the function f(x, y) with respect to x: $$f_x = \frac{\partial f}{\partial x} = \frac{\partial \cos{xy}}{\partial x}$$ Using the chain rule: $$f_x = -\sin{xy} \cdot \frac{\partial (xy)}{\partial x} = -y\sin{xy}$$

Step 2: Find partial derivative with respect to y (f_y)

Find the partial derivative of the function f(x, y) with respect to y: $$f_y = \frac{\partial f}{\partial y} = \frac{\partial \cos{xy}}{\partial y}$$ Using the chain rule: $$f_y = -\sin{xy} \cdot \frac{\partial (xy)}{\partial y} = -x\sin{xy}$$

Step 3: Find second partial derivative with respect to x (f_xx)

Now find the second partial derivative of f(x, y) with respect to x: $$f_{xx} = \frac{\partial^2 f}{\partial x^2} = \frac{\partial f_x}{\partial x} = \frac{\partial(-y\sin{xy})}{\partial x}$$ Using the chain rule: $$f_{xx} = -y^2\cos{xy}$$

Step 4: Find second partial derivative with respect to y (f_yy)

Next, find the second partial derivative of f(x, y) with respect to y: $$f_{yy} = \frac{\partial^2 f}{\partial y^2} = \frac{\partial f_y}{\partial y} = \frac{\partial(-x\sin{xy})}{\partial y}$$ Using the chain rule: $$f_{yy} = -x^2\cos{xy}$$

Step 5: Find mixed partial derivative with respect to x and y (f_xy)

Now, find the mixed second partial derivative of f(x, y) with respect to x and y: $$f_{xy} = \frac{\partial^2 f}{\partial x \partial y} = \frac{\partial f_x}{\partial y} = \frac{\partial(-y\sin{xy})}{\partial y}$$ Using the chain rule: $$f_{xy} = -x\cos{xy}$$

Step 6: Find mixed partial derivative with respect to y and x (f_yx)

Finally, find the mixed second partial derivative of f(x, y) with respect to y and x: $$f_{yx} = \frac{\partial^2 f}{\partial y \partial x} = \frac{\partial f_y}{\partial x} = \frac{\partial(-x\sin{xy})}{\partial x}$$ Using the chain rule: $$f_{yx} = -y\cos{xy}$$ Thus, the four second partial derivatives are: 1. $$f_{xx} = -y^2\cos{xy}$$ 2. $$f_{yy} = -x^2\cos{xy}$$ 3. $$f_{xy} = -x\cos{xy}$$ 4. $$f_{yx} = -y\cos{xy}$$