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How do we differentiate y=a^x when 'a' is an non zero real number

Firstly we must change it into a form we can deal with. To do this we take the natural log (ln) of both sides.

ln(y)=ln(ax

ln(y)=x(ln(a))         using our rules of logs

From here we differentiate. The differential of ln(f(x)) is [(d/dx)f(x)]/f(x)

(dy/dx)/y=ln(a)            

differentiating from above rule and ln(a) is just a constant so d/dx xln(a)= ln(a)

dy/dx=yln(a)        times both sides by y

dy/dx=(ax)(ln(a))  

subbing in y=ato get dy/dx in terms of x

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