Problem 31 - Calculus: Early Transcendentals

Question

Find the derivative of the function using the definition of derivative. State the domain of the function and the domain of its derivative. \( f(x) = x^4 \)

Step-by-Step Solution

Answer

The derivative of \( f(x) = x^4 \) is \( f'(x) = 4x^3 \). Both the function and its derivative have the domain \( (-\infty, \infty) \).

Step 1: Write the Definition of the Derivative

To find the derivative of the function using the definition of derivative, recall that the derivative of a function \( f(x) \), denoted as \( f'(x) \), is defined as:\[ f'(x) = \lim_{{h \to 0}} \frac{{f(x+h) - f(x)}}{h} \]

Step 2: Substitute the Function into the Definition

Substitute \( f(x) = x^4 \) into the definition of the derivative:\[ f'(x) = \lim_{{h \to 0}} \frac{{(x+h)^4 - x^4}}{h} \]

Step 3: Expand \( (x+h)^4 \)

Expand \( (x+h)^4 \) using the binomial theorem:\[ (x+h)^4 = x^4 + 4x^3h + 6x^2h^2 + 4xh^3 + h^4 \]

Step 4: Substitute the Expansion Back into the Limit

Substitute the expanded expression back into the limit:\[ f'(x) = \lim_{{h \to 0}} \frac{{x^4 + 4x^3h + 6x^2h^2 + 4xh^3 + h^4 - x^4}}{h} \]

Step 5: Simplify the Expression

Cancel out \( x^4 \) and factor \( h \) from the numerator:\[ f'(x) = \lim_{{h \to 0}} \frac{{4x^3h + 6x^2h^2 + 4xh^3 + h^4}}{h} = \lim_{{h \to 0}} (4x^3 + 6x^2h + 4xh^2 + h^3) \]

Step 6: Evaluate the Limit

Evaluate the limit by letting \( h \to 0 \):\[ f'(x) = 4x^3 + 6x^2(0) + 4x(0)^2 + (0)^3 = 4x^3 \]

Step 7: Determine the Domain of the Function and its Derivative

The function \( f(x) = x^4 \) is defined for all real numbers, so its domain is \( (-\infty, \infty) \). The derivative \( f'(x) = 4x^3 \) is also defined for all real numbers since polynomial functions are defined for all real numbers. The domain of its derivative is \( (-\infty, \infty) \).

Key Concepts

Definition of Derivative

Derivatives are fundamental in calculus and relate to how a function changes over its domain. The derivative of a function essentially describes its rate of change at any given point. The formal definition of a derivative is given by:

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

This definition allows us to find the slope of the tangent line to the function at any specific point. It focuses on examining the behavior of a function as we make tiny changes \( h \) to the input \( x \). By approaching the limit as \( h \to 0 \), we can precisely calculate the instant rate of change, or the derivative, at \( x \). Knowing this lets us analyze how functions behave and predict future values.

Binomial Theorem

The Binomial Theorem is a powerful tool for expanding expressions of the form \((x + y)^n\). It provides a way to expand polynomials without having to multiply the terms repeatedly. For an expression like \((x+h)^4\), the theorem can be expressed as:

• \((x+h)^4 = x^4 + 4x^3h + 6x^2h^2 + 4xh^3 + h^4\)

The coefficients \(1, 4, 6, 4, 1\) correspond to the terms in the polynomial expansion, which can also be found in Pascal's Triangle. The Binomial Theorem is invaluable when finding derivatives using the definition of derivative.When we expand \((x+h)^4\), it helps us isolate the changes caused by \(h\), simplifying the process of finding the limit as \(h \to 0\). This use of the theorem streamlines the calculation of derivatives, making it more efficient.

Polynomial Functions

Polynomial functions like \(f(x) = x^4\) are an important class of functions in mathematics. These functions are generally of the form \(a_nx^n + a_{n-1}x^{n-1} + ... + a_1x + a_0\), where \(a_n, a_{n-1}, ..., a_0\) are constants, and \(n\) is a non-negative integer.

• They are continuous and differentiable over all real numbers.
• They include terms with non-negative integer powers of \(x\), and coefficients that are real numbers.

Polynomial functions have many properties that make them straightforward to work with:

• They behave predictably, with smooth and continuous graphs.
• The degree of the polynomial determines the general shape and the number of extrema (minima and maxima).

Finding derivatives of polynomial functions, like in the example given, is a common task, and polynomials are excellent candidates for learning derivative concepts due to their simplicity and clarity.

Domain of a Function

Understanding the domain of a function is crucial, as it tells us the set of all possible input values (\(x\)) for which the function is defined. For the function \(f(x) = x^4\), it can be defined for all real numbers:

• The domain is \((-\infty, \infty)\).

This is because you can plug any real number into \(x\) without encountering division by zero or a negative square root.Similarly, the derivative \(f'(x) = 4x^3\) is also defined for all real numbers, given that it results from differentiating a polynomial function. Polynomial derivatives generally maintain the same domain as their parent functions:

• The domain of the derivative is also \((-\infty, \infty)\).

By knowing the domain, you understand the scope within which the function and its derivative are valid, ensuring any calculations made are applicable across relevant values of \(x\).