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Differential calculus - derivatives
 Derivative of a composite exponential function

  We use the logarithmic differentiation to find derivative of a composite exponential function of the form,     y = uv   where 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.
 Solution:
 
 
 
 
 
 
 
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