Since this is a linear function whose graph has a constant slope of 7, its inverse function will have a constant slope of 1/7 (the graph of the inverse function, when thought of as a function of x, will be the reflection of the original graph across the line y=x, so its slope will be the reciprocal of the slope of the original line). That will be the constant value of the derivative of the inverse function.
You can also solve for the inverse function: y=f(x)=7x+6\Rightarrow 7x=y-6\Rightarrow x=f^{-1}(y)=\frac{1}{7}y-\frac{6}{7}, implying that dx/dy=(f^{-1})'(y)=1/7 (there's no need to "swap" the x and y here unless you want to graph both f and f^{-1} on the same set of axes with the same independent variable to see the reflection property mentioned in the first paragraph).
