We
use the logarithmic differentiation to find derivative of a composite
exponential function of the form, y
= uvwhere
u
and v
are functions of the variable x
and u
> 0.
By
taking logarithms of both sides of the given exponential expression we
obtain,
ln
y= v
ln
u.
Differentiating both sides of the above
equation
with respect to
x
Use of the logarithmic differentiation
Derivatives of composite functions examples
Example:Find
the derivative of the function
Solution:
by
differentiating both sides of the above equation we get
or
Example:Find
the derivative of the function
Solution:
by
differentiating both sides of the above equation we get
Example:Find
the derivative of the function
Solution:
by
differentiating both sides of the above equation we get
Derivatives of the hyperbolic functions
We
use the derivative of the exponential function and the chain rule to
determine the derivative of the hyperbolic
sine and the hyperbolic cosine functions.
We
find derivative of the
hyperbolic tangent and hyperbolic cotangent functions applying
the quotient rule.
Therefore,
derivatives of the hyperbolic functions are
Derivatives of inverse hyperbolic functions
We
use the derivative of the logarithmic function and the chain rule to
find the derivative of inverse hyperbolic functions.
We
use the same method to find
derivatives of other inverse hyperbolic functions,
thus
Derivative of the inverse function
If
given a function y =
f(x) the derivative of
which y'(x) is
not 0
then, the derivative of the
inverse function x
= f-1(y)
is
Example:Find
the derivative x'(y)
if the given function f(x)= x+ln x.