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) = xr where x > 0 and r Î R, we can write  f (x) = erlnx  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:
Logarithmic differentiation
The derivative of the logarithm of a function  yf (x) is called the logarithmic derivative of the function, thus
Therefore, the logarithmic derivative is the derivative of the logarithm of a given function.
Functions contents F