
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 x^{2
}+ y^{2} 
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 b^{2}x^{2
}+ a^{2}y^{2} = a^{2}b^{2}
with the center at the origin, at the point (x_{1},
y_{1}).

Solution:
Differentiating
the equation of the ellipse we get,


so
the slope 

of the line
tangent to the ellipse at the point (x_{1},
y_{1}). 

Therefore,
the equation of the line tangent to the ellipse
at the point (x_{1},
y_{1})
is 


Generalized
power rule 
If
given f (x)
= x^{r}
where x >
0
and r
Î
R,
we can write f
(x)
= e^{r}^{lnx}
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 y
= f (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 



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