Problem 24 - Single Variable Calculus

Question

Find the derivative of the function. \(f(x)=-\frac{1}{3}\left(x^{-3}-x^{6}\right)\)

Step-by-Step Solution

Answer

The derivative of the given function \(f(x) = -\frac{1}{3} \left( x^{-3} - x^6 \right)\) is \(f'(x) = x^{-4} + 2x^5\).

Step 1: Apply the Constant Multiple Rule

The constant multiple rule states that the derivative of a constant multiplied by a function is equal to the constant multiplied by the derivative of the function. Let u(x) = -1/3, so the function becomes: \(f(x) = u(x) \left( x^{-3} - x^6 \right)\) Let's find the derivative of both terms, \(x^{-3}\) and \(x^{6}\), separately.

Step 2: Apply the Power Rule to the First Term

The power rule states that if \(f(x) = x^n\), then \(f'(x) = nx^{n-1}\). Applying the power rule to the first term \(x^{-3}\), we get: \(\frac{d}{dx} \left( x^{-3} \right) = -3x^{-3-1} = -3x^{-4}\)

Step 3: Apply the Power Rule to the Second Term

Applying the power rule to the second term \(x^{6}\), we get: \(\frac{d}{dx} \left( x^{6} \right) = 6x^{6-1} = 6x^{5}\)

Step 4: Combine the Derivatives and Apply the Constant Multiple Rule

Now that we have the derivatives of both terms, we can combine them and apply the constant multiple rule: \(\frac{d}{dx} \left[ u(x) \left( x^{-3} - x^6 \right) \right] = u(x) \left[ -3x^{-4} - 6x^5 \right] = -\frac{1}{3} \left[ -3x^{-4} - 6x^5 \right]\)

Step 5: Simplify the Expression

Simplify the expression to obtain the final derivative: \(f'(x) = -\frac{1}{3}\left( -3x^{-4} - 6x^5 \right) = x^{-4} + 2x^5\) The derivative of the given function \(f(x) = -\frac{1}{3} \left( x^{-3} - x^6 \right)\) is \(f'(x) = x^{-4} + 2x^5\).

Key Concepts

Constant Multiple Rule

The constant multiple rule is an essential tool when working with derivatives. It simplifies the process of taking derivatives for functions that are multiplied by a constant. This rule states that if you have a function, say, \(f(x)\), and it's multiplied by a constant value \(c\), then the derivative of this product is simply the constant multiplied by the derivative of the function. Make sure to:

• Identify the constant outside of the function. In this exercise, \(-\frac{1}{3}\) is the constant.
• Find the derivative of the function without the constant first. Break down into smaller parts if needed, then multiply the final derivative by the constant.

Utilizing this rule effectively can make deriving complex functions manageable by focusing on the function's main components first, and then adjusting for the constant.

Power Rule

The power rule is pivotal when finding the derivatives of polynomial terms in a function. It asserts that if you have a term in the form of \(x^n\), the derivative of this term will be \(nx^{n-1}\). This means:

• Multiply the exponent by the current coefficient of \(x\).
• Subtract one from the exponent.

Using the provided example, the derivative of \(x^{-3}\) becomes \(-3x^{-4}\), and \(x^6\) becomes \(6x^5\). The ability to apply the power rule straightforwardly aids in systematically deriving each term and streamlining the evaluation process, especially in polynomials.

Polynomial Functions

Polynomial functions are made up of terms that are constants, variables, or products of variables raised to whole number powers. They are easy to recognize and contain functions like \(x^6 - x^{-3}\), as seen in the given exercise. When deriving polynomial functions:

• Treat each term individually using the power rule.
• Remember the derivative of a constant is zero.
• Simplify by combining like terms if necessary.

In the exercise, each polynomial term's derivative is initially considered separately before applying the constant multiple rule.
Polynomials create a base framework for more complex functions, making these rules versatile in calculus. Working through the rules and encapsulating the knowledge of derivatives promotes comprehension of more advanced mathematical concepts.