Find (∂u / ∂y)x at the point (u, v)=(√(2), 1), if x=u^2+v^2 and y=u v .

Name: Problem 6, Chapter 14: Calculus
Uploaded: 2020-11-28T17:03:32-08:00
Duration: 3 min 39 s
Channel: Subham Jyoti Mishra
Description: Find (∂u / ∂y)x at the point (u, v)=(√(2), 1), if x=u^2+v^2 and y=u v .

step 1 So given question we have x is equal to u square plus v square and y is equal to uv we have to f

Find (∂u / ∂y)x at the point (u, v)=(√(2), 1), if x=u^2+v^2...

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Key Concepts

Chain Rule in Multivariable Calculus

The chain rule extends to functions of multiple variables and allows the computation of derivatives for composite functions. It is especially useful when variables are interdependent, as it links the rates of change between them through their functional relationships.

Partial Differentiation

In multivariable calculus, partial differentiation involves taking the derivative of a function with respect to one variable while holding the other variables constant. This concept is crucial for analyzing how a function changes along one dimension when the other dimensions remain fixed.

Implicit Differentiation

Implicit differentiation is a technique used when a function is defined indirectly by an equation involving several variables, rather than being given in an explicit form. By differentiating both sides of the relation with respect to the variable of interest and applying the necessary derivative rules, one can solve for the desired derivative.

Implicit Function Theorem

The implicit function theorem provides conditions under which a relation defined by an equation in several variables can be locally solved for one variable as a function of the others. This theorem not only guarantees the existence of such a function but also facilitates the computation of its derivative via implicit differentiation.

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