### 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(9)\cdot 9^x#

\[\begin{array}{rcl}

f'(x)&=&\displaystyle \frac{\dd}{\dd x} f(x)\\

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

&=& \displaystyle\frac{\dd}{\dd x} 9^x\\

&&\blue{\text{substituted }f(x)}\\

&=& \displaystyle\ln(9)\cdot 9^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} 9^x\\

&&\blue{\text{substituted }f(x)}\\

&=& \displaystyle\ln(9)\cdot 9^x\\

&&\blue{\text{applied exponential rule}}\end{array}\]

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