Differentiation: The derivative of standard functions
The derivative of exponential functions and logarithms
Now that we know the natural logarithm, we can introduce the derivative of exponential functions.
Exponential rule
For #\orange{a} \gt 0#
\[\dfrac{\dd}{\dd x}(\orange{a}^\blue{x})=\ln(\orange{a})\cdot \orange{a}^\blue{x}\]
Example
\[\begin{array}{rcl}\dfrac{\dd}{\dd x}(\orange{3}^\blue{x})&=&\ln(\orange{3})\cdot \orange{3}^\blue{x}\end{array}\]
We can now also calculate the derivative of the logarithm.
Logarithm rule
For #\orange{a} \gt 0#
\[\dfrac{\dd}{\dd x}(\log_\orange{a}(\blue{x}))=\dfrac{1}{\blue{x}\cdot\ln(\orange{a})}\]
Example
\[\begin{array}{rcl}\dfrac{\dd}{\dd x}(\log_{\orange{10}}(\blue{x}))&=&\dfrac{1}{\blue{x}\cdot\ln(\orange{10})}\end{array}\]
#f'(x)=# #\ln(5)\cdot 5^x#
\[\begin{array}{rcl}
f'(x)&=&\displaystyle \frac{\dd}{\dd x} f(x)\\
&&\blue{\text{definition derivative}}\\
&=& \displaystyle\frac{\dd}{\dd x} 5^x\\
&&\blue{\text{substituted }f(x)}\\
&=& \displaystyle\ln(5)\cdot 5^x\\
&&\blue{\text{applied exponential rule}}\end{array}\]
\[\begin{array}{rcl}
f'(x)&=&\displaystyle \frac{\dd}{\dd x} f(x)\\
&&\blue{\text{definition derivative}}\\
&=& \displaystyle\frac{\dd}{\dd x} 5^x\\
&&\blue{\text{substituted }f(x)}\\
&=& \displaystyle\ln(5)\cdot 5^x\\
&&\blue{\text{applied exponential rule}}\end{array}\]
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