Analysis of the expression:
The given expression is x log7(logx). To differentiate this expression with respect to x, we need to apply the product rule of differentiation. The product rule states that for two functions u(x) and v(x), the derivative of their product is given by u'(x)v(x) + u(x)v'(x).

Applying the product rule:
Let u(x) = x and v(x) = log7(logx).
Then, differentiate u(x) with respect to x: u'(x) = 1
Differentiate v(x) with respect to x: v'(x) = 1/(x log(7) x)
Now, applying the product rule, the derivative of the given expression is:
x * 1/(x log(7) x) + log7(logx) * 1
Simplifying the above expression, we get:
1/log(7) x + log7(logx)
Therefore, the derivative of x log7(logx) with respect to x is 1/log(7) x + log7(logx).