Problem 20 - 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. $$y=3 \sqrt[3]{x}$$

Step-by-Step Solution

Answer

\(y' = \frac{1}{3\sqrt[3]{x^2}}\)

Step 1: Rewrite the Function

The given function is written using a radical sign. Rewrite the power function with a fractional exponent to make differentiation easier. In this case, rewrite the cube root of \(x\) as \(x^{1/3}\). Therefore, the function becomes \(y=3x^{1/3}\).

Step 2: Differentiate using the Power Rule

Using the power rule for differentiation, which states that the derivative of \(x^n\) with respect to \(x\) is \(nx^{n-1}\), differentiate the function. Apply the rule to \(3x^{1/3}\) to get the derivative \(y' = 3*(1/3)x^{(1/3)-1} = x^{-2/3}\). Finally, multiply the constants to get \(y' = x^{-2/3}\).

Step 3: Simplify the Derivative

Simplify the derivative expression. For \(x^{-2/3}\), rewrite it in radical form to get \(y' = \frac{1}{3\sqrt[3]{x^2}}\).

Step 4: Check the Result with a Calculator (Optional)

Use a calculator with differentiation capability to verify that the result obtained in Step 3 is correct. Input the original function \(3\sqrt[3]{x}\) into the calculator and check if the displayed derivative matches the simplified derivative \(\frac{1}{3\sqrt[3]{x^2}}\).

Key Concepts

Fractional Exponents Differentiation

Differentiating functions with fractional exponents can seem daunting at first, but it follows a straightforward rule, much like differentiating integer power functions. When you come across a function with a fractional exponent like the cube root of a variable (expressed as \(x^{1/3}\), for instance), you can apply differentiation rules just as you would for any other power function.

The crucial step here is to rewrite the radical expression as a power with a fraction. Once you've rewritten \(3 \sqrt[3]{x}\) as \(3x^{1/3}\), the path is clear for standard differentiation techniques. This process not only helps in understanding the underlying operations but also in improving your ability to manipulate expressions and solve calculus problems more efficiently.

Remember that the fractional exponent corresponds to the root (the denominator of the exponent) and the power (the numerator of the exponent). So, for any radical, transforming it into an exponent will yield expressions that are simpler to differentiate using the power rule.

Power Rule for Differentiation

The power rule for differentiation is a core concept in calculus and it's your go-to tool for dealing with functions like \(x^n\), where \(n\) is any real number. This rule states that the derivative of \(x^n\) is \(nx^{n-1}\). When applying this to fractional exponents, it works the same way as with integers.

To put it into practice with your function \(3x^{1/3}\), you multiply the exponent by the coefficient (1/3 * 3), then subtract one from the exponent to get \(x^{(1/3)-1}\) or \(x^{-2/3}\). It may feel a bit tricky when the exponents are fractions, but with practice, it becomes second nature. Using the power rule correctly is a significant part of simplifying derivatives and making them easier to interpret and apply in further calculations or real-world contexts.

Simplifying Derivatives

Once you've found the derivative of a function, like our \(x^{-2/3}\) from the example, simplifying the expression is the next step to make it more usable. Simplification might involve converting negative exponents back to radical form or reducing complex fractions.

For instance, the derivative you obtained, \(x^{-2/3}\), can be rewritten as \(\frac{1}{3\sqrt[3]{x^2}}\). This reformulation turns the derivative into a format that is often easier to work with, especially for further integration or when evaluating the derivative at specific points. Plus, seeing your derivative in radical form can be handy when graphing the function or comparing it to other forms.

Simplification is not just cosmetic; it aids in computational efficiency and can reveal insights about the behavior of the function that may not be immediately obvious when looking at the derivative in its 'raw' form. As you delve into calculus, becoming comfortable with simplifying derivatives quickly and effectively will vastly improve your mathematical toolkit.