# Maclaurin Series of a^x

In this tutorial we shall derive the series expansion of the trigonometric function $${a^x}$$ by using Maclaurin’s series expansion function.

Consider the function of the form

\[f\left( x \right) = {a^x}\]

Using $$x = 0$$, the given equation function becomes

\[f\left( 0 \right) = {a^x} = {a^0} = 1\]

Now taking the derivatives of the given function and using $$x = 0$$, we have

\[\begin{gathered} f’\left( x \right) = {a^x}\ln a,\,\,\,\,\,\,\,\,\,\,f’\left( 0 \right) = {a^0}\ln a = \ln a \\ f”\left( x \right) = {a^x}{\left( {\ln a} \right)^2},\,\,\,\,\,\,\,\,\,\,f”\left( 0 \right) = {a^0}{\left( {\ln a} \right)^2} = {\left( {\ln a} \right)^2} \\ f”’\left( x \right) = {a^x}{\left( {\ln a} \right)^3},\,\,\,\,\,\,\,\,\,\,f”’\left( 0 \right) = {a^0}{\left( {\ln a} \right)^3} = {\left( {\ln a} \right)^3} \\ {f^{\left( {{\text{iv}}} \right)}}\left( x \right) = {a^x}{\left( {\ln a} \right)^4},\,\,\,\,\,\,\,\,\,\,{f^{\left( {{\text{iv}}} \right)}}\left( 0 \right) = {a^0}{\left( {\ln a} \right)^4} = {\left( {\ln a} \right)^4} \\ \cdots \cdots \cdots \; \cdots \cdots \cdots \cdots \cdots \cdots \cdots \cdots \cdots \cdots \cdots \cdots \cdots \\ \cdots \cdots \cdots \cdots \cdots \cdots \cdots \cdots \cdots \cdots \cdots \cdots \cdots \cdots \cdots \cdots \\ \end{gathered} \]

Now using Maclaurin’s series expansion function, we have

\[f\left( x \right) = f\left( 0 \right) + xf’\left( 0 \right) + \frac{{{x^2}}}{{2!}}f”\left( 0 \right) + \frac{{{x^3}}}{{3!}}f”’\left( 0 \right) + \frac{{{x^4}}}{{4!}}{f^{\left( {{\text{iv}}} \right)}}\left( 0 \right) + \cdots \]

Putting the values in the above series, we have

\[\begin{gathered} {a^x} = 1 + x\left( {\ln a} \right) + \frac{{{x^2}}}{{2!}}{\left( {\ln a} \right)^2} + \frac{{{x^3}}}{{3!}}{\left( {\ln a} \right)^3} + \frac{{{x^4}}}{{4!}}{\left( {\ln a} \right)^4} + \cdots \\ {a^x} = 1 + x\left( {\ln a} \right) + \frac{{{{\left( {x\ln a} \right)}^2}}}{{2!}} + \frac{{{{\left( {x\ln a} \right)}^3}}}{{3!}} + \frac{{{{\left( {x\ln a} \right)}^4}}}{{4!}} + \cdots \\ \end{gathered} \]

Kalani Jonathan

June 8@ 12:23 pmThis was good. I would like more information on the applications of these series. I am a teacher who would like to see the applicability of these series in science and every day life. Kindly assist me.

Tywad

June 2@ 3:13 amDepending on what level of maths you do teach- you could do a strange exercise asking whether 1^x can equal 2. Then when you see that there is some complex x where 1^x=2 you can point to here that exponentiation is not simply multiplying by itself a number of times.

Regarding real applications you see these sums used a lot in classical mechanics regarding oscillations, and stuff similar to above is also explored in quantum physics too.