The chain rule applications
      Implicit differentiation
         Implicit differentiation examples
      Generalized power rule
         Generalized power rule examples
Implicit differentiation
Let given a function  F = [y (x)]n,  to differentiate F we use the power rule and the chain rule,
which, for example, for n = 2 gives 
The same method we use to differentiate an equation of a curve given in the implicit form
F(x, y) = c  or  F(x, y) = 0.
So, the chain rule applied to differentiate the above equation with respect to x gives
Meaning, differentiate the equation with respect to both variables at the same time such that, when differentiating with respect to x, consider  y constant, while when differentiating with respect to y, consider x constant and multiply by  y'.
Implicit differentiation examples
Example:   Calculate the derivative y' of the equation of the curve  x2 + y2 - 2axy = 0.
Solution:   We use the above method of implicit differentiation to find y' of the equation,
Example:   Let determine the equation of the line tangent to the ellipse b2x2 + a2y2 = a2b2 with the center at the origin, at the point (x1, y1).
Solution:   Differentiating the equation of the ellipse we get,
so the slope   of the line tangent to the ellipse at the point (x1, y1).
Therefore, the equation of the line tangent to the ellipse at the point (x1, y1) is
Generalized power rule
If given  f (x) = x r where x > 0 and r R, we can write  f (x) = e r ln x  and apply the chain rule, then 
For
Example:   Find the derivative of the function  
Solution:   
Example:   Find the derivative of the function  
Solution:   
Example:   Find the derivative of the function  
Solution:   
Calculus contents C
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