Problem 21 - 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. $$y=-17 \sqrt{x^{3}}$$

Step-by-Step Solution

Answer

\(\frac{dy}{dx} = -\frac{51}{2} \sqrt{x}\)

Step 1: Express the function with a fractional exponent

Rewrite the power function with a fractional exponent to make differentiation straightforward. The square root of a number to a power can be expressed as that number to the power of half the original exponent. Therefore, \(y = -17 \sqrt{x^{3}}\) can be rewritten as \(y = -17x^{\frac{3}{2}}\).

Step 2: Differentiate the function

Use the power rule of differentiation which states that if \(y = ax^n\), then \(\frac{dy}{dx} = n \times ax^{n-1}\). Apply this rule to the given function. Differentiating \(-17x^{\frac{3}{2}}\) with respect to \(x\) gives us \(\frac{dy}{dx} = -17 \times \frac{3}{2} \times x^{\frac{3}{2}-1}\).

Step 3: Simplify the derivative

Simplify the expression from Step 2. The derivative of the function is \(\frac{dy}{dx} = -17 \times \frac{3}{2} \times x^{\frac{1}{2}}\) or \(\frac{dy}{dx} = -\frac{51}{2} \sqrt{x}\).

Step 4: Verification by calculator (if required)

Use a scientific calculator or a differentiation tool to calculate the derivative of the function and compare the result with the derived solution.

Key Concepts

Fractional Exponents

Understanding fractional exponents is essential for dealing with power functions in calculus. A fractional exponent indicates that an expression involves both a power and a root. For instance, expressing a square root (\(\sqrt{x}\)) is the same as raising a number to the power of 0.5. Generalizing this, if we have \(x^{a/b}\), it represents the \(b\)th root of \(x\) raised to the \(a\)th power.

When given a function like \(-17 \sqrt{x^{3}}\), converting it to a fractional exponent \(-17x^{\frac{3}{2}}\) simplifies the differentiation process. This conversion is based on the property that \(\sqrt[n]{x^{m}} = x^{\frac{m}{n}}\). In this case, we have a cube root inside a square root, so we multiply the exponents (3 * 1/2), resulting in a fractional exponent of 3/2.

Power Rule of Differentiation

The power rule is a fundamental technique for differentiating power functions with exponents. The rule states that if you have a function of the form \(y = ax^n\), where \(a\) is a constant and \(n\) is a real number, the derivative of \(y\) with respect to \(x\) is \(\frac{dy}{dx} = n \times ax^{n-1}\).

Applying this rule to our function with a fractional exponent, \(-17x^{\frac{3}{2}}\), tells us that each factor in the function is operated on by the rule. Multiplying the exponent by the coefficient and then subtracting 1 from the exponent gives us the simplified derivative, \(-17 \times \frac{3}{2} \times x^{\frac{1}{2}}\), which is \(-\frac{51}{2}\sqrt{x}\).

Simplifying Derivatives

After applying differentiation rules, it's crucial to simplify the resulting expression to make it more understandable and easier to evaluate. Start by multiplying the constants, and then apply any needed algebraic simplifications.

In the example, the derivative after applying the power rule is \(-17 \times \frac{3}{2} \times x^{\frac{1}{2}}\). By simplifying, we combine the constant factors, yielding \(-\frac{51}{2}\) times the square root of \(x\), which is our final derivative.

Simplification not only makes the derivative cleaner, but also prepares the function for further operations, such as evaluating the derivative at a specific value of \(x\) or integrating the function.