Problem 13
Question
Find the first partial derivatives of the following functions. $$g(x, y)=\cos 2 x y$$
Step-by-Step Solution
Verified Answer
The first partial derivatives of the function \(g(x, y) = \cos(2xy)\) are:
1. With respect to x: \(\frac{\partial g(x, y)}{\partial x} = -2y\sin(2xy)\)
2. With respect to y: \(\frac{\partial g(x, y)}{\partial y} = -2x\sin(2xy)\)
1Step 1: Differentiate g(x, y) with respect to x
To find the partial derivative of \(g(x, y)\) with respect to \(x\), we'll keep \(y\) constant and differentiate \(g(x,y)\) with respect to \(x\). Using the chain rule, we find:
$$\frac{\partial g(x,y)}{\partial x} = \frac{d (\cos(2xy))}{d (2xy)} \cdot \frac{\partial (2xy)}{\partial x}$$
2Step 2: Calculate the derivatives and find the partial derivative with respect to x
Now we calculate the respective derivatives:
$$\frac{d (\cos(2xy))}{d (2xy)} = -\sin(2xy)$$
and
$$\frac{\partial (2xy)}{\partial x} = 2y$$
Multiplying these derivatives together, we find the partial derivative with respect to \(x\):
$$\frac{\partial g(x, y)}{\partial x} = -2y\sin(2xy)$$
3Step 3: Differentiate g(x, y) with respect to y
To find the partial derivative of \(g(x, y)\) with respect to \(y\), we'll keep \(x\) constant and differentiate \(g(x, y)\) with respect to \(y\). Using the chain rule, we find:
$$\frac{\partial g(x, y)}{\partial y} = \frac{d(\cos(2xy))}{d(2xy)} \cdot \frac{\partial(2xy)}{\partial y}$$
4Step 4: Calculate the derivatives and find the partial derivative with respect to y
Now we calculate the respective derivatives:
$$\frac{d (\cos(2xy))}{d (2xy)} = -\sin(2xy)$$
and
$$\frac{\partial (2xy)}{\partial y} = 2x$$
Multiplying these derivatives together, we find the partial derivative with respect to \(y\):
$$\frac{\partial g(x, y)}{\partial y} = -2x\sin(2xy)$$
5Step 5: Write down the first partial derivatives
The first partial derivatives of the function \(g(x, y)=\cos(2xy)\) are:
$$\frac{\partial g(x, y)}{\partial x} = -2y\sin(2xy)$$
and
$$\frac{\partial g(x, y)}{\partial y} = -2x\sin(2xy)$$
Other exercises in this chapter
Problem 13
Find the domain of the following functions. $$f(x, y)=\sqrt{25-x^{2}-y^{2}}.$$
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Find all critical points of the following functions. $$f(x, y)=x^{3} / 3-y^{3} / 3+3 x y$$
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