Find the derivative at $x=2$ of the function $f(x)=3 x$.

Step 1: Recall the definition of the derivative

The derivative of a function $f(x)$ at a point $x$ is defined by the limit of the difference quotient. This formula allows us to find the instantaneous rate of change of the function.

Step 2: Substitute the function into the definition

We substitute the given function $f(x) = 3x$ into the derivative definition. This means replacing $f(x+h)$ with $3(x+h)$ and $f(x)$ with $3x$.

Step 3: Simplify the expression

Next, we expand the term $3(x+h)$ to $3x + 3h$. The $3x$ terms cancel out, leaving us with $3h$ in the numerator.

Step 4: Evaluate the limit

Since $h \neq 0$ in the limit process, we can cancel out $h$ from the numerator and denominator. This simplifies the expression to a constant, 3. The limit of a constant is the constant itself.

Step 5: Find the derivative at $x=2$

The derivative $f'(x) = 3$ is a constant, meaning the rate of change of the function $f(x)=3x$ is always 3, regardless of the value of $x$. Therefore, at $x=2$, the derivative is also 3.

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