Question

Consider a function \(f(x)\) and its derivative \(d f l d x\). Does this derivative have to be a function of \(x\) ?

Short answer

Answer: The derivative of a function is usually related to the independent variable x, and it will be a function of x if it is defined for all values of x in the domain of the original function. However, there could be cases where the derivative might not exist for certain values of x, making it a partial function of x instead.

Step-by-step solution

Recall the definition of a derivative

A derivative of a function, \(f(x)\), is a measure of the rate of change of the function with respect to the independent variable \(x\). The derivative is denoted by \(\frac{d f}{d x}\) or \(f'(x)\). It is defined as the limit: $$ \frac{d f}{d x} = \lim_{h \to 0} \frac{f(x + h) - f(x)}{h} $$ where \(h\) is a very small change in the independent variable \(x\).

Analyze the relationship between the function and its derivative

The derivative of a function is itself a function if it is defined for all values of \(x\) in the domain of \(f(x)\). However, there could be cases where the derivative might exist for some values of \(x\) but not the others. In those cases, the derivative will not be a function of \(x\) in the traditional sense but still can be considered a partial function.

Provide examples

(a) Consider the function \(f(x) = x^2\). Its derivative is given by: $$ \frac{d f}{d x} = 2x $$ This derivative is a function of \(x\), as it is defined for all values of \(x\). (b) Consider the function \(g(x) = |x|\) where \(|x|\) denotes the absolute value of \(x\). This function looks like a "V" shape with a sharp point at \(x=0\). At the point \(x=0\), the function does not have a well-defined tangent, and therefore, the derivative does not exist at \(x=0\). In this case, the derivative of \(g(x)\) exists for all \(x\) except \(x=0\), so it can be considered a partial function of \(x\).

Conclusion

The derivative of a function, \(f(x)\), will be a function of \(x\) if it is defined for all values of \(x\) in the domain of \(f(x)\). In some cases, the derivative might not exist for certain values of \(x\), making it a partial function of \(x\). However, in general, the derivative can be considered related to the variable \(x\).

Key concepts

Derivative

A derivative is a mathematical concept that represents how a function changes as its input changes. In simple terms, it measures the function's sensitivity to changes in its input variable, often denoted as \(x\). The derivative is commonly written as \(\frac{d f}{d x}\) or \(f'(x)\). It is calculated using the limit:

• \(\frac{d f}{d x} = \lim_{h \to 0} \frac{f(x + h) - f(x)}{h}\)

This formula helps us understand how the function \(f(x)\) behaves in an infinitesimally small neighborhood around \(x\). For example, if \(f(x) = x^2\), then its derivative, \(\frac{d f}{d x} = 2x\), indicates that the function's rate of change increases linearly with \(x\).
This is crucial for various applications, from physics to economics, where we need to understand dynamics and change.

Function

A function is a fundamental concept in calculus and mathematics in general. It connects each element of one set, known as the domain, with exactly one element of another set, called the codomain. Functions are usually denoted as \(f(x)\), indicating that \(f\) is dependent on the variable \(x\).
Functions can take many forms, from simple linear functions like \(f(x) = mx + c\) to more complex ones such as trigonometric functions. The idea is that for every input \(x\), there is a unique output \(f(x)\).
In studying derivatives, understanding the original function is vital since the derivative depends on how that function behaves.

• If you know \(f(x) = x^2\), knowing the structure helps find \(f'(x)\).
• Knowing how functions behave aids in predicting their derivatives.

Rate of Change

The rate of change tells us how one quantity changes in relation to another. In mathematical terms, the derivative of a function at a given point represents the rate of change of the function’s value with respect to its input variable. It's essentially the slope of the function's graph at that point.
Consider \(f(x) = x^2\) with a derivative of \(2x\). This tells us that:

• As \(x\) increases, \(f(x)\) increases more quickly.
• The slope of the tangent line to the curve at any \(x\) gives the instantaneous rate of change.

This concept is essential for analyzing motion in physics, finding optimal solutions in economics, and more. Understanding how quickly change occurs helps in predicting future behavior of systems.

Differentiable Functions

A differentiable function is a function that has a derivative at every point in its domain. This means that it has a well-defined tangent and, therefore, a predictable rate of change. Differentiability is an important property as it ensures that functions behave nicely.
If a function like \(f(x) = x^2\) is differentiable everywhere, its derivative function \(f'(x) = 2x\) is well-defined for all \(x\).
However, not all functions are differentiable at every point. Consider the function \(g(x) = |x|\). It is not differentiable at \(x = 0\) due to the sharp turn, which means the derivative does not exist there.

• This lack of differentiability at points is what creates discontinuities in the derivative.
• Identifying differentiable functions is crucial for ensuring accurate predictions and analyses.