Notes on Derivatives: Product Rule, Notation, and Applications

Product Rule and Notation: Key Points

The derivative of a product is not simply the derivative of one term times the other; it requires the product rule which yields two terms.
In the left-hand limit for a product, you effectively see two terms, illustrating why the product rule is necessary (as opposed to the sum of differences being obviously simpler).
When differentiating a factored expression, it helps to clearly identify the first factor and the second factor; doing this with dx/dx notation often clarifies which factor is being differentiated.
Notation matters: many people prefer the prime notation f' for derivatives, but the d/dx notation remains popular because it makes it clearer which factor’s derivative is being taken in a product.
The product rule is not primarily about “computing quickly” but about rewriting the derivative of a product in terms of derivatives of the factors.
The power rule is dedicated to computing derivatives of monomials like x^n; the product rule is used when you have a product of functions, e.g., x^2 sin(x).

Power rule: differentiate a single term like x^n o n x^{n-1}.
Product rule: differentiate a product of two functions, using the two-factor formula to account for the derivative of each factor:
\frac{d}{dx}[u(x)v(x)] = u'(x)\,v(x) + u(x)\,v'(x).
In practice, you may prefer to apply the rule step-by-step (two separate thoughts) rather than compressing everything into one step.
Temptation to combine steps is common, but separating the derivative of each factor helps maintain two distinct lines of thought and reduces mistakes.

Example: if f(x) = x^2 \sin x, then
\frac{d}{dx}[x^2 \sin x] = 2x \sin x + x^2 \cos x.
This demonstrates the product rule in action and how the result is a sum of terms, each involving the derivative of one factor multiplied by the other.

In quotient-rule problems, you may be asked to simplify your result.
Important distinction: you cannot cancel terms inside a numerator unless they are actual common factors in numerator and denominator.
You can cancel factors that appear as common factors in numerator and denominator, but you cannot cancel a whole term that is not a common factor.
The line from the transcript: “Nothing cancels. You cannot cancel terms. You can cancel factors. 3x − 1 is not a factor of the numerator. It is a factor of one term of the numerator.”
Conceptual takeaway: treat cancellation as factoring, not as canceling arbitrary portions of an expression.

If you are given a position function, such as s(t) (or x(t), y(t), or h(t)), typical questions involve rates of change:
- Velocity: v(t) = \frac{ds}{dt} = s'(t) or v(t) = \frac{dx}{dt} depending on the variable.
- Acceleration: a(t) = \frac{dv}{dt} = \frac{d^2 s}{dt^2} = \frac{d^2 x}{dt^2}.
A common scenario discussed: throwing a ball in the air or moving distance horizontally, where you consider how speed and position change over time.
If speed is increasing, you are changing the rate of position; in other words, you are considering derivatives of derivatives.

Since velocity is the rate of change of position, acceleration is the rate of change of velocity; this explains why two derivatives are often involved when analyzing motion.
The transcript notes examples where derivatives appear in real-world contexts beyond math class:
- Economics: marginal cost is a derivative, i.e., MC(q) = C'(q) where C(q) is total cost.
- Chemistry: reaction rates are derivatives of concentration with respect to time, e.g., \frac{d[A]}{dt} for a reactant A.
- Biology: blood flow rates can be described via derivatives with respect to time.

Notation clarity (d/dx vs prime) matters for problem solving, especially when differentiating products.
The product rule is a fundamental tool that converts a derivative of a product into a sum involving derivatives of the individual factors.
It’s helpful to keep separate lines of thinking when applying the product rule rather than trying to compress everything into a single step.
In applications, derivatives model rates of change across disciplines, reinforcing the unifying idea of the derivative as a rate.

Product rule: \frac{d}{dx}[u(x)v(x)] = u'(x)\,v(x) + u(x)\,v'(x).
Example: \frac{d}{dx}[x^2 \sin x] = 2x \sin x + x^2 \cos x.
Position/velocity/acceleration:
- v(t) = \frac{dx}{dt}
- a(t) = \frac{dv}{dt} = \frac{d^2 x}{dt^2}
Cancellation rule: cancel common factors, not arbitrary terms; e.g., a common factor a in numerator and denominator can cancel, but a term that is not a factor cannot.
Real-world links: marginal cost, reaction rates, blood flow rates.