Differentiation of Exponential and Logarithmic Functions, Derivative of Logarithmic Function (Continued), Second Order Derivatives

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Derivative of Logarithmic Function (Continued):

  • We know that derivative of the function w.r.t. x is , where n is a constant. This rule is not applicable, when exponent is a variable. In such cases we take logarithm of the function and then find its derivative.

  • Therefore, this process is useful, when the given function is of the type . For example, etc.

  • Note: Here may be constant.

Derivative of w.r.t.

Let

Taking log on both sides, we get

or

Or

Thus,

Example:

If , prove that

Solution:

It is given that

Taking logarithm on both sides, we get

Or

Or

Taking derivative with respect to on both sides of (2), we get

Second Order Derivatives:

  • In the previous lesson we found the derivatives of second order of trigonometric and inverse trigonometric functions by using the formulae for the derivatives of trigonometric and inverse trigonometric functions, various laws of derivatives, including chain rule, and power rule discussed earlier in lesson 21.

  • In a similar manner, we will discuss second order derivative of exponential and logarithmic functions:

Example:

If , show that

Solution:

We have,

{Using (1)}

Or

Taking derivative of both sides of (2), we get

Or [Dividing throughout by ]

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