Question

If the limit definition of a derivative can be used to find \(f^{\prime},\) then what is the purpose of using other rules to find \(f^{\prime} ?\)

Short answer

Answer: Using rules like the power rule, chain rule, product rule, and quotient rule is advantageous because they simplify the process and save time when finding the derivative of a function. While the limit definition can always be used, it can be computationally intensive and difficult to apply, especially for complex functions. These rules offer a faster and more efficient way to find the derivative, making it easier to work with various types of functions and apply them in different situations.

Step-by-step solution

Understanding the Limit Definition of Derivative

The limit definition of a derivative is given by: \[f'(x) = \lim_{h\to0} \frac{f(x+h) - f(x)}{h}\] Using this definition, we can find the derivative of a function by substituting the function into the formula and finding the limit as \(h\) approaches 0.

Introduction to Other Rules for Finding Derivatives

Apart from the limit definition, there are other rules to find the derivative of a function: 1. Power Rule: If \(f(x) = x^n\), then \(f'(x) = nx^{n-1}\) 2. Chain Rule: If \(f(x) = g(h(x))\), then \(f'(x) = g'(h(x)) \cdot h'(x)\) 3. Product Rule: If \(f(x) = g(x)h(x)\), then \(f'(x) = g'(x)h(x) + g(x)h'(x)\) 4. Quotient Rule: If \(f(x) = \frac{g(x)}{h(x)}\), then \(f'(x) = \frac{g'(x)h(x) - g(x)h'(x)}{h^2(x)}\)

Purpose of Using Other Rules

The purpose of using other rules for finding derivatives is to simplify the process and save time. While the limit definition always works, it can be computationally intensive and difficult to use, especially for complex functions. The other rules offer a faster and more efficient way to find the derivative of a function, making it easier for us to work with various types of functions and apply them in different situations.

Comparing Methods

To demonstrate the difference between using the limit definition and other rules, let's consider the following function: \[f(x) = x^3\] Using the limit definition of a derivative, we have: \[f'(x) = \lim_{h \to 0} \frac{(x + h)^3 - x^3}{h}\] Solving this limit can be cumbersome and requires simplification and substitution. On the other hand, using the power rule: \[f'(x) = 3x^2\] We can directly and easily find the derivative without going through the limit process. This illustrates the purpose and advantage of using other rules to find derivatives.

Key concepts

Limit Definition of a Derivative

The limit definition of a derivative is fundamental to calculus and serves as the bedrock for understanding how derivatives are computed. It's written as:
\[\begin{equation}\begin{aligned}f'(x) = \lim_{{h\to0}} \frac{f(x+h) - f(x)}{h}\end{aligned}\end{equation}\]This expression tells us that the derivative of a function at a certain point is the slope of the tangent line to the function at that point. To find this slope, we consider the rate at which the function values change as the input values become infinitesimally close to each other.While the limit definition is precise, it can be quite complex and time-consuming to apply, especially for intricate functions. This complexity is a key reason alternative rules for differentiation were developed.

Power Rule

The power rule dramatically simplifies the process of differentiation when dealing with polynomial functions. It states that for a function \[\begin{equation}\begin{aligned}f(x) = x^n,\end{aligned}\end{equation}\]the derivative is\[\begin{equation}\begin{aligned}f'(x) = nx^{n-1}.\end{aligned}\end{equation}\]Not only is this method less computationally intensive than the limit definition, but it also allows for immediate computation of the derivative. This is incredibly useful when dealing with higher powers of x, as calculating limits can become unwieldy.

Chain Rule

The chain rule is essential when tackling the derivatives of composite functions—functions that are composed of one function inside another. It is expressed as:
\[\begin{equation}\begin{aligned}\text{If } f(x) = g(h(x)), \text{ then } f'(x) = g'(h(x)) \cdot h'(x).\end{aligned}\end{equation}\]This rule enables us to differentiate the outer function (often referred to as the 'outer layer') first and then multiply it by the derivative of the inner function ('inner layer'). By breaking down complex functions into simpler components, we make the differentiation process more digestible and straightforward.

Product Rule

Whenever you encounter a function that is the product of two other differentiable functions, you'll need the product rule to find the derivative. The formula is given by:
\[\begin{equation}\begin{aligned}\text{If } f(x) = g(x)h(x), \text{ then } f'(x) = g'(x)h(x) + g(x)h'(x).\end{aligned}\end{equation}\]This allows you to differentiate each function separately and then combine them. It's crucial to respect the order since you first multiply the derivative of the first function by the second function, and then add the product of the first function and the derivative of the second function. The product rule helps in avoiding the lengthy process of using the limit definition for each term.

Quotient Rule

Similar to the product rule, the quotient rule is designed for functions that are ratios of two other differentiable functions. The quotient rule is a bit more involved and is written as:\[\begin{equation}\begin{aligned}\text{If } f(x) = \frac{g(x)}{h(x)}, \text{ then } f'(x) = \frac{g'(x)h(x) - g(x)h'(x)}{h^2(x)}.\end{aligned}\end{equation}\]Using this rule, one would take the derivative of the numerator and multiply it by the denominator, subtract the product of the numerator and the derivative of the denominator, and then divide the entire result by the square of the denominator. It avoids the convoluted limit calculations that would otherwise be necessary for such functions.