Question

For the constant function \(f(x)=c,\) use the definition of the derivative to show that \(f^{\prime}(x)=0\).

Short answer

Answer: The derivative of a constant function is 0.

Step-by-step solution

Write down the definition of the derivative

Our starting point is the definition of the derivative, which states that: $$f^{\prime}(x) = \lim_{h \to 0} \frac{f(x + h) - f(x)}{h}$$

Substitute the function values for the constant function

Since our function \(f(x) = c\), we can substitute the function values for both \(f(x + h)\) and \(f(x)\) as follows: $$f^{\prime}(x) = \lim_{h \to 0} \frac{c - c}{h}$$

Simplify the expression

We can now simplify the expression inside the limit by subtracting the two constants c: $$f^{\prime}(x) = \lim_{h \to 0} \frac{0}{h}$$

Evaluate the limit

Since dividing 0 by any number results in 0, we can see that: $$f^{\prime}(x) = \lim_{h \to 0} 0 = 0$$ Thus, we have shown that the derivative of the constant function \(f(x) = c\) is \(f^{\prime}(x) = 0\).

Key concepts

Limit Definition of Derivative

Understanding the limit definition of the derivative is crucial for analyzing changes in mathematical functions. It is formally articulated as the limit of the ratio of the change in the function value to the change in the function's input as the latter approaches zero:
\[f^{\text{'}}(x) = \text{lim}_{{h \to 0}} \frac{{f(x + h) - f(x)}}{h}\]
When considering the derivative, you are essentially trying to find the instantaneous rate of change at a specific point on a graph of the function. For example, in physics, the derivative of the position function with respect to time gives the velocity – the rate at which the position is changing over time. The use of limits ensures that the derivative describes this rate of change precisely at a point, rather than over an interval, capturing the basic notion of an instantaneous rate of change.

Constant Function Properties

A constant function is arguably one of the simplest forms of mathematical functions, yet it holds important properties. A function of the form
\(f(x) = c\)
where \(c\) is a constant, is called a constant function because for all input values \(x\), the output will always be \(c\). It doesn't matter what the input value is, the output remains unchanged. This means the graph of a constant function is a horizontal line in the coordinate plane. Since there is no change in the output, the rate of change, which is what a derivative measures, is zero. Hence, constant functions have a derivative of zero.

Evaluating Limits

Evaluating limits is a process central to calculus, often used to understand the behavior of functions as inputs approach a particular value. It involves finding the value that a function approaches as the input approaches some value. For example, when dealing with the limit
\[ \text{lim}_{{h \to 0}} \frac{{f(x + h) - f(x)}}{h} \],
it is essential to determine the value this fraction approaches as \(h\) goes to 0. If the function's value becomes closer and closer to a fixed number as \(h\) becomes small, then that number is the limit of the function. In some cases, as with the constant function, the limit can be evaluated directly because the expression simplifies to a constant value.

Simplifying Mathematical Expressions

Simplifying mathematical expressions is an invaluable skill, making complex calculus operations more manageable. When solving problems using the limit definition of the derivative, you often encounter expressions that might initially seem complicated. It is essential to simplify these expressions to perform limit evaluation. Subtracting constants, factorizing, and canceling out terms are standard simplification techniques. For a constant function, simplification shows the change in function value to be zero, leading directly to the simple conclusion that the derivative is also zero. Simplification not only aids in clearer understanding but also reduces the potential for computational errors.