Differentiation: The derivative of standard functions
The base e and the natural logarithm
The number #\e\approx 2.71828182846\ldots# is an important number in mathematics and therefore has its own name.
The derivative of #\orange{\e}^\blue{x}# is equal to itself:
\[\dfrac{\dd}{\dd x}\orange{\e}^\blue{x} =\orange{\e}^\blue{x}\]
For #\orange{\e}# we use the calculation rules for exponents.
Example
\[\begin{array}{rcl} \dfrac{\dd}{\dd x}(8 \, \orange{\e}^\blue{x})&=& 8\cdot \dfrac{\dd}{\dd x}\orange{\e}^\blue{x}\\ &=& 8 \, \orange{\e}^\blue{x}\end{array}\]
We can take the number #e# as the base number of a logarithm. We call this the natural logarithm and note it with #\ln#.
The natural logarithm
The natural logarithm is
\[\ln(\blue{x})=\log_\orange{\e}(\blue{x})\]
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.
Example
\[\begin{array}{rcl}\\ \ln(\orange{\e}^\blue{x})&=&\blue{x}\\ \end{array}\]
\[\begin{array}{rcl}
f'(x)&=&\dfrac{\dd}{\dd x} 9\cdot \e^{x}\\
&&\blue{\text{definition derivative}}\\
&=&9\cdot \dfrac{\dd}{\dd x} \left( \e^x\right)\\
&&\blue{\text{constant rule}}\\
&=&9\cdot \e^{x}\\
&&\blue{\dfrac{\dd}{\dd x}\left(\e^x\right)=\e^x}
\end{array}\]
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