### Differentiation: The derivative of standard functions

### The base e and the natural logarithm (revisited)

We have introduced *Euler's number*, #\e\approx 2.71828182846\ldots#, in the chapter on exponential functions. This important number has a unique property.

The derivative of #\orange{\e}^\blue{x}# is equal to itself:

\[\dfrac{\dd}{\dd x}\orange{\e}^\blue{x} =\orange{\e}^\blue{x}\]

**Example**

\[\begin{array}{rcl} \dfrac{\dd}{\dd x}(8 \, \orange{\e}^\blue{x})&=& 8\cdot \dfrac{\dd}{\dd x}\orange{\e}^\blue{x}\\ &=& 8 \, \orange{\e}^\blue{x}\end{array}\]

As we have seen before, for #\orange{\e}# we use the *calculation rules for exponents*.

Recall, that when we take the number #e# as the base number of a logarithm, we call it the natural logarithm and note it with #\ln#.

The natural logarithm

The **natural logarithm **is

\[\ln(\blue{x})=\log_\orange{\e}(\blue{x})\]

**Example**

\[\begin{array}{rcl}\\ \ln(\orange{\e}^\blue{x})&=&\blue{x}\\ \end{array}\]

With the natural logarithm we use the same *rules* as with other logarithms. *Rewriting* to different base numbers goes the same as with the natural logarithm.

\[\begin{array}{rcl}

f'(x)&=&\dfrac{\dd}{\dd x} 9\cdot \e^{x}\\

&&\blue{\text{definition derivative}}\\

&=&9\cdot \dfrac{\dd}{\dd x} \left( \e^x\right)\\

&&\blue{\text{constant rule}}\\

&=&9\cdot \e^{x}\\

&&\blue{\dfrac{\dd}{\dd x}\left(\e^x\right)=\e^x}

\end{array}\]

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