Calculus question! (d)/(dx) [((1)/(5x))^(4x)]

Question

Calculus question!
`(d)/(dx) [((1)/(5x))^(4x)]`

Answer 1

First, some rewriting:


`(\mathrm d)/(\mathrm dx)(5x)^(-4x)=(\mathrm d)/(\mathrm dx)\exp\left(\ln(5x)^(-4x)\right)=(\mathrm d)/(\mathrm dx)\exp\left(-4x\ln(5x)\right)`

Now taking the derivative is just a matter of applying the chain rule. Since
`(\mathrm d)/(\mathrm dx)e^(f(x))=(\mathrm df(x))/(\mathrm dx)e^(f(x))`, you end up with


`(\mathrm d)/(\mathrm dx)(5x)^(-4x)=(\mathrm d)/(\mathrm dx)[-4x\ln(5x)]\exp\left(-4x\ln(5x)\right)=(\mathrm d)/(\mathrm dx)[-4x\ln(5x)](5x)^(-4x)`

The product rule tells you that


`(\mathrm d)/(\mathrm dx)[-4x\ln(5x)]=-4x(\mathrm d)/(\mathrm dx)[\ln(5x)]+\ln(5x)(\mathrm d)/(\mathrm dx)[-4x]=-4x*\frac5{5x}-4\ln(5x)=-4(1+\ln(5x))`

So the derivative of the original function is


`(\mathrm d)/(\mathrm dx)(5x)^(-4x)=-4(1+\ln(5x))(5x)^(-4x)`