Derivative - Chain Rule

Category: Differential Calculus, Algebra

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Given the following functions:

                      

                    
Find dy/dx in terms of x and y. 

Solution:  

The first thing that we have to do is to get the derivative of each functions with respect to their independent variables. 

For the first function

For the second function

For the third function

Therefore,

The above equation is called the Derivative by Chain Rule. As you noticed that the differentials like du and dv will be cancelled. Multiply the derivative of three functions, we have 

The final answer must be free from other variables like u and v. We have to eliminate u and v by substituting their values with x. We know that 

Therefore,