Compute all the first and second partial derivatives of the function. We assume that y>0𝑦>0 .f(x,y)=xy−3x−ln(y6) .
Question
Compute all the first and second partial derivatives of the function. We assume that y>0𝑦>0 .f(x,y)=xy−3x−ln(y6) .
Solution
First, let's compute the first partial derivatives:
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Partial derivative with respect to x (∂f/∂x):
∂f/∂x = y - 3
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Partial derivative with respect to y (∂f/∂y):
∂f/∂y = x - 6/y
Now, let's compute the second partial derivatives:
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Second partial derivative with respect to x (∂²f/∂x²):
∂²f/∂x² = 0
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Second partial derivative with respect to y (∂²f/∂y²):
∂²f/∂y² = 6/y²
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Mixed second partial derivative with respect to x and y (∂²f/∂x∂y):
∂²f/∂x∂y = 1
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Mixed second partial derivative with respect to y and x (∂²f/∂y∂x):
∂²f/∂y∂x = 1
So, the first and second partial derivatives of the function f(x,y)=xy−3x−ln(y6) are as above.
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