91Ó°ÊÓ

The Product Rule is a fundamental concept in calculus, used to find the derivative of a product of two functions. If you have a function that is the product of two other functions, say \(u(x)\) and \(v(x)\), then the Product Rule provides a way to find the derivative of this product. Imagine you have \(f(x) = u(x)\cdot v(x)\). To get the derivative, you would use the formula:\[\frac{d}{dx}f(x) = u(x) \cdot \frac{d}{dx}v(x) + v(x) \cdot \frac{d}{dx}u(x)\]This formula combines the derivatives of \(u(x)\) and \(v(x)\) by multiplying each function by the derivative of the other and summing the results.
Consider the function from our exercise, \(f(x) = (x-1)(3x+4)\). Applying the Product Rule:

• Identify \(u(x) = x - 1\) and \(v(x) = 3x + 4\).
• Find their derivatives: \(\frac{d}{dx}u(x) = 1\) and \(\frac{d}{dx}v(x) = 3\).
• Substitute into the Product Rule formula: \(u(x)\,d(v) + v(x)\,d(u) = (x-1) \cdot 3 + (3x+4) \cdot 1\).
• Finally, simplify to get \(6x + 1\).

By following these steps, you can efficiently find the derivative of any product of two functions.

The Quotient Rule is a method for differentiating functions that are in the form of one function divided by another, symbolized as \( \frac{u(x)}{v(x)} \). It is crucial when simplifying such functions before differentiation isn't enough or when a derivative is needed quickly.
The formula for the Quotient Rule is:\[\frac{d}{dx}\left(\frac{u(x)}{v(x)}\right) = \frac{v(x)\cdot \frac{d}{dx}u(x) - u(x)\cdot \frac{d}{dx}v(x)}{(v(x))^2}\]This formula might look intimidating, but it's just carefully applying the derivatives of the numerator and the denominator:

• Differentiate the numerator \(u(x)\).
• Differentiate the denominator \(v(x)\).
• Multiply the derivative of the denominator by the numerator function \(u(x)\), then subtract from it the product of the derivative of the numerator by the denominator.
• Finally, divide the entire expression by the square of the denominator \(v(x)^2\).

This process provides a systematic way to find the derivatives of functions that are quots, making it a powerful tool for calculus students.

Function expansion is a technique used to rewrite a function in a simpler form that makes it easier to take its derivative, integrate, or simplify further.
In the exercise, we were asked to expand the function \((x-1)(3x+4)\). This involves distributive property: multiplying each term in one polynomial by each term of the other polynomial.
Steps for expanding \((x-1)(3x+4)\):

• First, distribute \((x - 1)\) across every term in \((3x + 4)\).
• Calculate \(3x \cdot x\) and \( 3x \cdot (-1)\) to get \(3x^2 - 3x\).
• Next, do \(4 \cdot x\) and \(4 \cdot (-1)\) to get \(4x - 4\).
• Combine all terms to form the expanded function \(3x^2 + x - 4\).

Once expanded, finding the derivative becomes straightforward, using basic rules of differentiation on each term individually.