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4.12 Derivatives of Exponential and Logarithmic Functions

The number ‘ e ’

Therefore the limit of , as n approaches infinity is

The number defined by , or by the sum of the convergent series , is denoted by ‘ e ’. Its value, correct to 6 decimal places is 2 . 718282 i.e. clearly 2 < e < 3. It is also a fundamental constant like p and it is the base of the natural or Napierian, or hyperbolic, logarithms.

The exponential function ex which is sometimes written as

e x p x is

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Another important function is

* Exponential functions in which a variable quantity occurs as an index, or exponent.

The function are examples of exponential functions.

Now , let y = f ( x ) = ex then f ( x + h ) = e x + h = ex . eh

Then by the 1st principal of derivatives we get

Differentiation of

Differentiation of ‘log x’:-

Let y = logyex i.e. f ( x ) = log x Þ f ( x + h ) = log ( x + h )

Index

4. 1 Derivability At A Point
4. 2 Derivability In An Interval
4. 3 Derivability And Continuity Of A Function At A Point
4. 4 Some Counter Examples

4. 5 Interpretation Of Derivatives
4. 6 Theorems On Derivatives (differentiation Rules)
4. 7 Derivatives Of Standard Functions
4. 8 Derivative Of A Composite Function
4. 9 Differentiation Of Implicit Functions

4.10 Derivative Of An Inverse Function
4.11 Derivatives Of Inverse Trigonometric Functions
4.12 Derivatives Of Exponential & Logarithmic Functions
4.13 Logarithmic Differentiation
4.14 Derivatives Of Functions In Parametric Form
4.15 Higher order Derivatives

Chapter 5