Question

Question #b31ec

Answer 1

`f'(x)=(1+ln(x)-xln(5)ln(x))/5^x`

Explanation:

This problem can still be simplified for easier differentiating.

Using the Change of Base Formula:

`log_5(x)=ln(x)/ln(5)`

This simplifies to:
`(x(ln5)(lnx/ln5))/5^x=(xlnx)/5^x`

Applying the Quotient Rule:

`f'(x)=((d/dx[xlnx])5^x-(d/dx[5^x])xlnx)/(5^x)^2`
`f'(x)=([(x)(1/x)+1(lnx)]5^x-(5^xln5)(xlnx))/5^(2x)`
`f'(x)=(1+ln(x)-xln(5)ln(x))/5^x`