Question

How do I find the derivative of a fraction raised to a power

Answer 1

Let
`b>0` be any real number with
`b\\eq e`. Then you can write


`b^x=e^(\ln b^x)=e^(x\ln b)`

Now, by the chain rule for differentiation, the derivative is


`(\mathrm d)/(\mathrm dx)e^(x\ln b)=e^(x\ln b)(\mathrm d)/(\mathrm dx)(x\ln b)=\ln b~e^(x\ln b)=\ln b~b^x`