Derivative rules: Constant rule, is it 0 or 1?

Question

Because of the constant rule, $\frac{d}{dx}(k) = 0$, where k is any constant.

However in one of the practice problems: Differentiate integer powers (mixed positive and negative), I tried to apply the same and $\frac{d}{dx}(x) = 1$ instead of zero.

Can someone explain to me why it is 1 instead of 0 here?

Answer 1

The definition of a derivative here is: $n \cdot x ^{n-1}$.

Example: $f(x) = x^2$ $$\frac{d}{dx}(x^2), n=2 \\ \text{applying the definition of the derivative} \\ n \cdot x ^{n-1} = 2x^{2-1} = 2x^1 = 2x \\ \text{Now apply this rule to the variable in your question} \\ \frac{d}{dx}(x), \text{where $x = x^1$} \\ n = 1, n \cdot x ^{n-1} = 1 \cdot x ^0 = 1$$

The main point, $x$ is a variable. If $x$ was defined as a constant than it would be $0$.

Answer 2

A constant function $f(x) = c$ means at any value of $x$, $f(x) = c$. $F(x)=x$ is a linear not a constant function.

Answer 3

$$\frac d{dx}(x) = \frac d{dx}(x^1) = 1 \cdot x^{1-1} = x^0 = 1.$$