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How do you take the derivative of a^x ?

There are two ways you can take the derivative of a^x.

1) 

Let y = a^x    now we're trying to find dy/dx

2)

ln(y) = xln(a)    take logs of both sides and use log rules

3)

(dy/dx)*(1/y) = ln(a)   take the derivative of both sides using the chain rule                                           on the left hand side.

4)

dy/dx = ln(a)*y              multiply both sides by y

5)

dy/dx =  ln(a) *a^x        realise y= a^x and replace it

Now we're done!

Alternatively we could realise that any exponent can be written as e to the power of something with a log in it.

So

1)

y = a^x = (e^ln(a))^x    just rewritting 'a'

2)

y = e^xln(a)                multiplying exponent rule

3)

dy/dx = ln(a)*e^xln(a)   take the derivative of both sides using the chain                                                   rule for the right hand side

4)

dy/dx  = ln(a)*a^x                         substitute back to get desired result

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