Problem 30 - Brief Applied Calculus

Question

Find the derivative of each function in two ways: a. Using the Quotient rule. b. Simplifying the original function and using the Power Rule. Your answers to parts (a) and (b) should agree. $$ \frac{1}{x^{4}} $$

Step-by-Step Solution

Answer

The derivative of \( \frac{1}{x^4} \) is \(-4x^{-5}\), using both methods.

Step 1: Identify the Function for Differentiation

First, let's rewrite the function to make differentiation easier. The function given is \( f(x) = \frac{1}{x^4} \), which can be rewritten as \( f(x) = x^{-4} \).

Step 2: Using the Quotient Rule

The quotient rule states that if \( f(x) = \frac{u(x)}{v(x)} \), then \( f'(x) = \frac{v(x)u'(x) - u(x)v'(x)}{v(x)^2} \). For the function \( \frac{1}{x^4} \), we have \( u(x) = 1 \) and \( v(x) = x^4 \). Calculate derivative as:\[ u'(x) = 0, \quad v'(x) = 4x^3 \]Thus, using the quotient rule:\[ f'(x) = \frac{x^4 \cdot 0 - 1 \cdot 4x^3}{(x^4)^2} = \frac{-4x^3}{x^8} = -4x^{-5} \]

Step 3: Simplify Function for Power Rule Application

We already rewrote \( f(x) = x^{-4} \). Now, apply the power rule for derivatives, which is \( \frac{d}{dx}x^n = nx^{n-1} \).

Step 4: Using the Power Rule

Using the power rule on \( f(x) = x^{-4} \), we differentiate:\[ f'(x) = -4(x^{-4-1}) = -4x^{-5} \]

Step 5: Verify Consistency of Results

Both methods give the same result for the derivative: \( f'(x) = -4x^{-5} \). Thus, the solutions are consistent with each other.

Key Concepts

Quotient Rule

The Quotient Rule is a handy method in calculus used to find the derivative of a function that is the division of two separate functions. To put it simply, if you have a function expressed as \( f(x) = \frac{u(x)}{v(x)} \), where \( u(x) \) and \( v(x) \) are two functions that you can differentiate separately, the derivative of \( f(x) \) is found using the formula:\[f'(x) = \frac{v(x)u'(x) - u(x)v'(x)}{(v(x))^2}\]This formula arises from applying the product rule and the chain rule. Let's break it down further:

• Compute the derivative of the numerator function \( u'(x) \).
• Compute the derivative of the denominator function \( v'(x) \).
• Multiply the original denominator function \( v(x) \) by the derivative of the numerator \( u'(x) \).
• Multiply the original numerator function \( u(x) \) by the derivative of the denominator \( v'(x) \).
• Subtract the second product from the first.
• Divide the resulting expression by the square of the original denominator \( (v(x))^2 \).

In our example, using the quotient rule allows us to find the derivative of \( \frac{1}{x^4} \) by recognizing \( u(x) = 1 \) and \( v(x) = x^4 \). The result is \( -4x^{-5} \).

Power Rule

The Power Rule is one of the most straightforward and frequently used rules when applying calculus to find derivatives. It states that if you have a function \( y = x^n \), its derivative \( y' \) is \( nx^{n-1} \). This means you bring the power down in front as a multiplier and reduce the power by one.

• Identify the exponent \( n \).
• Multiply the entire function by the exponent \( n \).
• Reduce the exponent by one.

Applying this to \( x^{-4} \), the derivative is calculated as follows:- The current power is \(-4\).- Multiply by \(-4\) to get \(-4x^{\_}\).- The new power becomes \(-4 - 1 = -5\), giving \(-4x^{-5}\) as the derivative.This approach is particularly simple when the function is given in the form of a power, making the calculation quick and direct.

Simplifying Expressions

Simplifying mathematical expressions is an essential skill in calculus, as it often leads to easier and more straightforward differentiation. By rewriting or breaking down functions, you can apply basic derivative rules like the power rule more efficiently. In the case of \( \frac{1}{x^4} \), simplifying this expression helps reveal the form needed for direct application of the power rule. We transformed:

• \( \frac{1}{x^4} \) into \( x^{-4} \).

This transformation is crucial. It converts a division-based function into a simple power function, making it eligible for the straightforward application of derivative rules like the power rule.Simplifying expressions before differentiation not only reduces chances of error but also often aligns results through different methods, as seen by the consistency in deriving \(-4x^{-5}\) by both quotient and power rule in this problem.