Question

Find an expression for
d^n y/dx^n if y=a^x

Answer 1

Use the exponential and logarithm functions to rewrite

`y=a^x=e^(\ln a^x)=e^(x\ln a)`

Then by the chain rule,

`(\mathrm dy)/(\mathrm dx)=e^(x\ln a)(\mathrm d(x\ln a))/(\mathrm dx)=e^(x\ln a)\ln a=a^x\ln a`

`(\mathrm d^2y)/(\mathrm dx)=e^(x\ln a)\ln a(\mathrm d(x\ln a))/(\mathrm dx)=e^(x\ln a)(\ln a)^2=a^x(\ln a)^2`

and so on, with

`(\mathrm d^ny)/(\mathrm dx^n)=a^x(\ln a)^n`