Why is derivative of $x$ with respect to $x$ equal to $1$?

Question

I just started learning the derivatives of inverse function. The first example is based on the fact that $\frac{\mathsf{d}x}{\mathsf{d}x} = 1$ and it is stated that I should know this already. However I don't recall this fact, moreover I don't know how you can solve $\frac{\mathsf{d}x}{\mathsf{d}x}$.

Could somebody explain me how do we solve such question and is this rule apply to every function?

Answer 1

Maybe it becomes clear if you write it as $$\frac{\mathrm d}{\mathrm dx} x = (x)' = 1$$ This is basically the first derivative you learn of.

Answer 2

The derivative is defined as

$$\lim_{h \rightarrow 0}\frac{f(x+h)-f(x)}{h}$$ which in this case ($f(x)=x$) reduces to $$\lim_{h \rightarrow 0}\frac{x+h-x}{h}=\lim_{h \rightarrow 0}\frac{h}{h}=1$$

Hope that helps, otherwise let me know in the comments below.