Problem 12 - 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 Negative Exponent. $$y=2 x^{-3}$$

Step-by-Step Solution

Answer

The derivative of the function y = 2*x^(-3) is -6*x^(-4), which can also be written as -6/x^4.

Step 1: Identify the Function Type

Recognize that the given function, which is of the form y = ax^n, where a is a constant and n is a negative integer, is a power function with a negative exponent.

Step 2: Apply the Power Rule

To find the derivative of the power function, use the power rule for differentiation, which states that the derivative of x^n is n*x^(n-1). In this case, the derivative of 2*x^(-3) will be -3*2*x^(-3-1).

Step 3: Simplify the Derivative

Multiply the constants and subtract one from the exponent to simplify the expression: -3*2*x^(-3-1) becomes -6*x^(-4).

Step 4: Write the Final Answer

Express the final derivative in the simplest form: the derivative of y = 2*x^(-3) can be written as -6/x^4.

Key Concepts

Power Rule for Differentiation

Understanding the power rule for differentiation is crucial when dealing with polynomials and functions involving exponents. In its simplest form, the power rule states that if you have a function of the form f(x) = x^n, where n is any real number, then the derivative of f with respect to x is f'(x) = nx^{n-1}. Let's see how this applies when n is negative.

For our exercise example, the function in question is y = 2x^{-3}. Here, n = -3. Following the power rule, we differentiate x^{-3} by multiplying by the exponent and then decreasing the exponent by one, which gives us -3x^{-4}. Don't forget to also multiply by the constant in front of the x, so we include the 2 to get -6x^{-4} as the derivative.

Simplifying Derivatives

After applying the power rule, we often need to simplify the derivative to make the expression clearer and easier to work with. In our previous example involving the function y = 2x^{-3}, we found that the derivative is -6x^{-4}. To simplify, we can express negative exponents as fractions. For instance, x^{-4} simplifies to 1 / x^4. Thus, the simplified form of the derivative is -6 / x^4. Remember, simplifying the derivative can also assist with further calculations, like finding the slope of the tangent line at a particular point or solving for critical points in calculus.

It's important to note that the simplification process may vary when dealing with more complex functions or additional operations, such as products or quotients. Always aim to express your final answer in the most reduced form for clarity.

Negative Exponents in Calculus

In calculus, negative exponents play a significant role in differentiation and integration. A negative exponent indicates that the base of the term is in the denominator when written as a fraction. For example, x^{-n} is equivalent to 1 / x^n. When differentiating functions with negative exponents using the power rule, the result usually includes negative exponents as well.

However, it's often desirable to express your final derivative without negative exponents for better readability and to facilitate further algebraic manipulations. As seen in the earlier sections, we convert -6x^{-4} to -6 / x^4 by recognizing that the negative exponent indicates the reciprocal of the base raised to the positive exponent. Another key point is that negative exponents can imply decay in certain contexts, such as when modeling decreasing processes in physics or biology.