Question

Question #a787b

Answer 1

`d/dx(log(xe^x)/log(5))=(x+1)/(xlog(5))`

Explanation:

Another way to do this is to simplify at first using the rule `log(AB)=log(A)+log(B)`:

`log(xe^x)/log(5)=(log(x)+log(e^x))/log(5)=(log(x)+x)/log(5)`

Don't forget that `log(5)` is a constant, so we can bring it out when differentiating:

`d/dx((log(x)+x)/log(5))=1/log(5)d/dx(log(x)+x)`

The derivative of `log(x)` is `d/dxlog(x)=1/x`. The derivative of `x` is `1`.

`1/log(5)d/dx(log(x)+x)=1/log(5)(1/x+1)=1/log(5)((1+x)/x)`

Combining this all:

`d/dx(log(xe^x)/log(5))=(x+1)/(xlog(5))`

Answer 2

`(1+x) / (xln5)`

Explanation:

STEP 1: Rewrite the expression using logarithm rules
If we remember our logarithm rules, we will recall that `log_bx = logx/logb`.
Using the logarithm rule above, we can rewrite `log(xe^x)/log5` as `log_5(xe^x)`.

STEP 2: Differentiate
Recall our differentiation rule for logarithms: `d/dx log_ax=1/(xlna)`

Combining our log derivative rule with the chain rule, we get:
`d/dx log_5(xe^x) = 1/((xe^x)ln5) * d/dx(xe^x)`
To take the derivative of `xe^x` we must remember to apply the product rule:
`1/((xe^x)ln5) * d/dx(xe^x) = 1/((xe^x)ln5) * [1*e^x + x*e^x]`

STEP 3: Simplify
`1/((xe^x)ln5) * [1*e^x + x*e^x]=[e^x + xe^x] / ((xe^x)ln5) = (1+x) / (xln5)`