Find the derivative at x=2 of the function f(x)=3x.
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Step-by-Step Solution
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=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.