91Ó°ÊÓ

The Quotient Rule is a vital technique in calculus for differentiating functions that are expressed as a quotient of two other functions (i.e., one function divided by another). When you're given a function in the form of \( f(x) = \frac{u(x)}{v(x)} \), the Quotient Rule helps in finding its derivative by employing the formula: \[ \left( \frac{u}{v} \right)' = \frac{u'v - uv'}{v^2} \]
Here's how it works:

• Differentiate the numerator: \(u'\).
• Differentiate the denominator: \(v'\).
• Substitute these derivatives into the formula, remembering to subtract the product \(uv'\) from \(u'v\).
• Divide the result by the square of the denominator, \(v^2\).

In the given problem, the function is \(\frac{2}{x^4}\). Here, \(u=2\) (a constant) and \(v=x^4\). Since the derivative of a constant \(u'\) is zero, we only needed to calculate \(-uv'\), leading to \(-\frac{16x^3}{x^8} = -\frac{16}{x^5}\). By applying the Quotient Rule correctly, we accurately attain the derivative of the rational component in the function.

The Power Rule is one of the simplest yet incredibly powerful tools in calculus for finding the derivative of functions featuring powers of \(x\). If your function is of the form \(x^n\), the derivative is given by: \[ \frac{d}{dx} x^n = nx^{n-1} \]
Follow these simple steps to use the Power Rule:

• Identify the exponent \(n\) of \(x\).
• Multiply \(x^n\) by the exponent \(n\).
• Decrease the exponent by one.

In our original problem, for the term \(-x^3\), the Power Rule gives us \(-3x^2\), since differentiating \(x^3\) results in \(3x^{2}\), and we retain the negative sign. Constants, like \(2\) here, derived to zero, simplifying our final derivative.

Function Differentiation is the process of finding the derivative of a function—essentially measuring how the function's output changes in response to small changes in the input. By using both the Quotient Rule and the Power Rule, we effectively differentiated the given function in pieces and then assembled to form the complete derivative.
The problem function was \( f(x) = \frac{2}{x^4} - x^3 + 2 \). By breaking it down:

• The \( \frac{2}{x^4} \) was differentiated using the Quotient Rule to get \(-\frac{16}{x^5}\).
• The \(-x^3\) was differentiated using the Power Rule to get \(-3x^2\).
• The derivative of the constant \(2\) was \(0\).

Bringing these together resulted in the derivative function \( f'(x) = -\frac{16}{x^5} - 3x^2 \). Recognizing and applying different differentiation techniques to individual function components is crucial for solving complex calculus problems.