Problem 19 - Technical Mathematics with Calculus

Question

Find the derivative of each function. Verify some of your results by calculator. As usual, the letters \(a, b, c, \ldots\) represent constants. Power Function with Fractional Exponent. $$y=4 \sqrt{x}$$

Step-by-Step Solution

Answer

The derivative of the function \(y = 4\sqrt{x}\) is \(y' = \frac{2}{\sqrt{x}}\).

Step 1: Express the function with a rational exponent

Rewrite the function using a fractional exponent. The square root of x can be expressed as x raised to the power of 1/2. So, the function: $$y = 4\sqrt{x}$$ can be rewritten as: $$y = 4x^{1/2}$$.

Step 2: Apply the power rule for differentiation

Use the power rule, which states that the derivative of x^n, denoted as d/dx of x^n, is n*x^(n-1). Apply this rule to the function: $$y' = 4 * \frac{1}{2} * x^{{1/2}-1} = 2x^{-1/2}$$.

Step 3: Simplify the expression

Convert the negative exponent into a radical expression to simplify the derivative: $$y' = 2x^{-1/2} = \frac{2}{\sqrt{x}}$$.

Key Concepts

Fractional Exponents

Understanding fractional exponents is essential when working with power functions, especially during differentiation. A fractional exponent represents a root along with an exponentiation. For instance, the square root of a number is the same as raising that number to the power of 1/2. This relationship can be generalized as: if you have an expression like \(a^{m/n}\), it is equivalent to the nth root of \(a\) raised to the mth power, or \(\sqrt[n]{a^m}\).

In the context of the exercise, we have a function \(y = 4\sqrt{x}\) which can be expressed with a fractional exponent as \(y = 4x^{1/2}\). This format is particularly useful for differentiation, as it fits directly into the rules we use to find derivatives, which leads us to our next concept, the power rule for differentiation.

Power Rule for Differentiation

The power rule is a basic yet powerful tool in calculus. It states that the derivative of a power function \(x^n\) is \(nx^{n-1}\). This makes calculating derivatives straightforward when dealing with polynomial functions. Applying the power rule to a fractional exponent stays the rule's course, only with the adjustment that the exponent is a fraction.

In our given problem, utilizing the power rule for the function \(y = 4x^{1/2}\), we multiply the exponent by the coefficient resulting in \(4 * \frac{1}{2}\) and reduce the exponent by one, yielding \(x^{1/2 - 1}\) or \(x^{-1/2}\), which becomes the simplified expression for the derivative: \(y' = 2x^{-1/2}\).

Simplifying Derivatives

After applying the differentiation rules, simplifying the derivative is crucial for easier interpretation and further applications, such as solving equations or evaluating limits. A negative exponent indicates division by that variable's corresponding positive exponent, transforming \(y' = 2x^{-1/2}\) into \(y' = \frac{2}{\sqrt{x}}\), a much more manageable form for further calculations or graphical representations.

While simplifying, it's also beneficial to look out for common factors that may be canceled out or expressions that can be factored to yield the simplest form of the derivative. This last step does not only make the derivative look neat, but it also aligns with the mathematical preference for expressions to be presented in their most reduced form whenever possible.