Derivative of f(x) = x via the power rule

Write $f(x) = x = x^1$. By the power rule, $\frac{d}{dx}[x^n] = n x^{n-1}$, so with $n=1$ we get $f'(x) = 1\cdot x^{1-1} = 1\cdot x^0 = 1\cdot 1 = 1$.

Since $f'(x) = 1$ for every value of $x$, the slope of the tangent line to $f(x) = x$ is the same, namely $1$, at every point on the graph. This makes sense because $f(x)=x$ is itself a straight line with slope $1$, so its tangent line at any point coincides with the line itself.