Find the derivative of 1/0.3.

Step-by-step solution

1. 1

   Simplify the expression first. Multiplying numerator and denominator by $10$ clears the decimal:

   $\frac{1}{0.3} = \frac{10}{3} = 3.\overline{3}$

2. 2

   Notice there is no variable. Neither $x$ nor any other variable appears, so $f(x) = \tfrac{10}{3}$ is a constant function — its value is the same for every input.

3. 3

   Apply the constant rule. For any constant $c$,

   $\frac{d}{dx}(c) = 0$

   This follows straight from the limit definition: $\lim_{h\to0}\tfrac{c - c}{h} = \lim_{h\to0}\tfrac{0}{h} = 0$.

4. 4

   Interpret geometrically. The graph of $y = \tfrac{10}{3}$ is a horizontal line. A horizontal line has slope $0$ at every point, and the derivative is exactly that slope.

5. 5

   Distinguish from a similar-looking expression. If instead the expression were $\tfrac{1}{0.3x}$, the variable would be present and the derivative would be $-\tfrac{1}{0.3x^2} = -\tfrac{10}{3x^2}$. The presence or absence of the variable is the whole question.