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How to Find Related Rates of a Two Variable Differentiable Function?

TopRelated rates of two different variables means both variables are related to each other. Here we will discuss how to find related rates of a two variable differentiable function. If an equation is given and there are two different variable x, y, they both are related to each other, in this case we can say that value of one variable depends on other variable. Let's take a scenario: Suppose a person wants to go from city 'a' to city 'b' and distance between both cities is 10 miles. Suppose at any time instant, person stands at place 'p', suppose distance between 'a' to 'p' is 'x' and 'p' to 'b' is 'y', and value of 'x' and 'y' vary with respect to time (t) . So in this case 'x' and 'y' are related to each other by equation x + y = 10, differentiation of this equation with respect to time’t’ gives: d(f) / d(t) = δ(f) / δ(x) * d(x) / d(t) + δ(f) / δ(y) * d(y) / d(t). There are some steps to find the related rates of two variables:-

First of all draw a picture, then find all variable in the given diagram. After this identify the variable whose rate of change we want to find. After this applies the formula which relates these variables, after this differentiate implicitly this equation and put all values to find other variable.
 
Example:- Find related rates of a two variable differentiable function in case of Right Triangle?
 
Solution) In this triangle 'x' and 'y' are related by pythagoras theorem, so relation is x2 + y2 = (5)2 ---> equation 1.
Here given that y = 3 then value of 'x' is 4, and given that dy/dt = -1, so differentiate this equation with respect to 't' we get: 2x.dx/dt + 2y.dy/dt = 0, now put the value of x ,y and dy/dt, we will get:
2. 4 dx/dt + 2. 3. (-1) = 0 => 8. dx/dt - 6 = 0 => dx /dt = 6/8 => ¾.