the process used to differentiate functions using x and y together
implicit differentiation
The process of differentiating functions with respect to both x and y variables together is called partial differentiation. In partial differentiation, we treat one variable as constant while differentiating with respect to the other variable. Consider the following function:
f(x,y) = 3x^2y + 2xy^2
To differentiate this function partially with respect to x, we hold y constant and differentiate only with respect to x:
∂f/∂x = 6xy + 2y^2
Similarly, to differentiate this function partially with respect to y, we hold x constant and differentiate only with respect to y:
∂f/∂y = 3x^2 + 4xy
It is important to note that the partial derivative of a function with respect to x or y is itself a function of x and y. These partial derivatives are computed using the same rules of differentiation as used for ordinary functions. The notation used for partial differentiation is different, as shown above, where the symbol ∂ is used instead of d to indicate partial differentiation.
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