how to calculate differetiation of implict function by partial differentiation?
Answers
Step-by-step explanation:
Sometimes a function of several variables cannot neatly be written with one of the variables isolated. For example, consider the following function x2y3z+cosycosz=x2cosxsiny. It would be practically impossibly to isolate z let alone any other variable. Fortunately, the concept of implicit differentiation for derivatives of single variable functions can be passed down to partial differentiation of functions of several variables.
Suppose that we wanted to find ∂z∂x. Then we would
take the partial derivatives with respect to x of both sides of this equation and isolate for ∂z∂x while treating y as a constant. Applying the product and chain rule where appropriate, we have that:
(1)
∂∂x(x2y3z+cosycosz)=∂∂x(x2cosxsiny)y3(2xz+x2∂z∂x)−cosysinz∂z∂x=(2xcosx−x2sinx)siny2xy3z+x2y3∂z∂x−cosysinz∂z∂x=(2xcosx−x2sinx)sinyx2y3∂z∂x−cosysinz∂z∂x=(2xcosx−x2sinx)siny−2xy3z∂z∂x(x2y3−cosysinz)=(2xcosx−x2sinx)siny−2xy3z∂z∂x=(2xcosx−x2sinx)siny−2xy3zx2y3−cosysinz
hope it is useful
mark my answer brainliest
dont forget to follow