Given the function f(x, y) = x2y4 − cos 2xy, find the partial derivatives
Question
Given the function f(x, y) = x2y4 − cos 2xy, find the partial derivatives
Solution
Sure, let's find the partial derivatives of the function f(x, y) = x^2*y^4 - cos(2xy).
- Partial derivative with respect to x (∂f/∂x):
The derivative of x^2y^4 with respect to x is 2xy^4.
The derivative of -cos(2xy) with respect to x is 2y*sin(2xy) (using the chain rule, where the derivative of cos(u) is -sin(u) and u = 2xy).
So, ∂f/∂x = 2xy^4 + 2ysin(2xy).
- Partial derivative with respect to y (∂f/∂y):
The derivative of x^2y^4 with respect to y is 4x^2y^3.
The derivative of -cos(2xy) with respect to y is 2x*sin(2xy) (again using the chain rule).
So, ∂f/∂y = 4x^2y^3 + 2xsin(2xy).
So the partial derivatives of the function f(x, y) = x^2y^4 - cos(2xy) are ∂f/∂x = 2xy^4 + 2ysin(2xy) and ∂f/∂y = 4x^2y^3 + 2x*sin(2xy).
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