Product Rule, Chain Rule & Inverse Functions
What happens when functions are multiplied together, nested inside each other, or run in reverse? In this lesson, you’ll unlock three powerful rules — the Product Rule, the Chain Rule, and the Inverse Function Derivative — that let you handle all of these situations. Once you have these tools, there’s almost no function you can’t differentiate!
Every time engineers design something, they combine simpler pieces:
- Product rule: A factory’s revenue is (price per unit) × (units sold). Both change over time — how fast does revenue change? You need the product rule!
- Chain rule: Your phone’s GPS converts satellite signals → position → speed. Each step feeds into the next, just like composite functions.
- Inverse functions: A thermometer converts temperature → mercury height. But you read it backward: mercury height → temperature. That’s an inverse function!
Topics Covered
- Quotient rule review: \(d\!\left(\frac{f}{g}\right) = \frac{g\,df - f\,dg}{g^2}\)
- Orders of infinitesimals: which tiny terms to keep and which to drop
- Differentiating \(\frac{f}{g^2}\) by extending the quotient rule
- Product rule: \(d(fg) = g\,df + f\,dg\) with geometric interpretation
- Generalized product rule for many factors (e.g., \(f^2 \cdot g^3 \cdot h^{-4}\))
- Composite functions and the chain rule
- Derivative of an inverse function: \(\frac{dx}{dy} = \frac{1}{dy/dx}\)
- Monotonicity and invertibility via the discriminant
Lecture Video
Key Frames from the Lecture
An infinitesimal \(dx\) is a quantity so tiny it’s essentially zero — but not exactly zero. We use it to measure how much a function changes when we nudge its input by a hair.
Key idea: if \(dx\) is infinitesimally small, then \(dx^2\) (i.e., \(dx \times dx\)) is overwhelmingly smaller. Think of it this way: if \(dx = 0.001\), then \(dx^2 = 0.000001\) — a thousand times smaller! So we can safely drop terms with \(dx^2\) or higher powers.
We already derived the quotient rule in a previous lesson. Here is the result:
\[d\!\left(\frac{f}{g}\right) = \frac{g\,df - f\,dg}{g^2}\]
Read it as: “bottom times change-in-top, minus top times change-in-bottom, all over bottom squared.”
Quotient Rule Review & Orders of Infinitesimals
When we derived the quotient rule, we expanded everything and then dropped the higher-order terms — the products like \(df \cdot dg\) that are infinitesimally small compared to \(df\) or \(dg\) alone.
The rule of thumb: always keep the lowest-order infinitesimal terms and throw away anything smaller.
| Term | Order | Keep or Drop? |
|---|---|---|
| \(df\) | 1st order | Keep |
| \(dg\) | 1st order | Keep |
| \(df \cdot dg\) | 2nd order | Drop |
| \(df^2\) | 2nd order | Drop |
If you have a sum like \(3\,df + 5\,dg + 2\,df\,dg\), the answer is just \(3\,df + 5\,dg\).
Extending the Quotient Rule: Differentiating \(\frac{f}{g^2}\)
What if the denominator is \(g^2\) instead of \(g\)? We can treat \(g^2\) as a single function and apply the quotient rule:
\[d\!\left(\frac{f}{g^2}\right) = \frac{g^2\,df - f\,d(g^2)}{g^4}\]
We need \(d(g^2)\). Expanding \((g + dg)^2 - g^2 = 2g\,dg + (dg)^2\), and dropping the \((dg)^2\) term:
\[d(g^2) = 2g\,dg\]
Substituting back and simplifying:
\[d\!\left(\frac{f}{g^2}\right) = \frac{g^2\,df - f \cdot 2g\,dg}{g^4} = \frac{g\,df - 2f\,dg}{g^3}\]
Explore the quotient rule — compare \(f/g\) and \(f/g^2\):
The Product Rule
To differentiate a product \(y = f \cdot g\), consider what happens when both \(f\) and \(g\) change by tiny amounts \(df\) and \(dg\):
\[(f + df)(g + dg) = fg + g\,df + f\,dg + df\,dg\]
Subtracting the original \(fg\) and dropping the tiny \(df\,dg\) term:
When two functions are multiplied together, the change in their product equals the first function times the change in the second, plus the second function times the change in the first.
\[\boxed{d(fg) = g\,df + f\,dg}\]
Geometric Interpretation
Think of \(f \cdot g\) as the area of a rectangle with sides \(f\) and \(g\).
When you stretch both sides a little, the new area gains:
- A thin horizontal strip: height \(f\), width \(dg\) → area \(f\,dg\)
- A thin vertical strip: width \(g\), height \(df\) → area \(g\,df\)
- A tiny corner square: \(df \times dg\) → negligible!
So the total change in area is \(g\,df + f\,dg\).
Explore the rectangle interpretation — drag the slider to change the “nudge”:
Generalized Product Rule
What if you have many factors multiplied together? The pattern extends naturally. To differentiate, go through each factor one at a time: differentiate that one factor, leave the rest alone, then add up all the pieces.
Example: \(y = f^2 \cdot g^3 \cdot h^{-4}\)
\[dy = \underbrace{2f\,df \cdot g^3 \cdot h^{-4}}_{\text{differentiate } f^2} + \underbrace{f^2 \cdot 3g^2\,dg \cdot h^{-4}}_{\text{differentiate } g^3} + \underbrace{f^2 \cdot g^3 \cdot (-4)h^{-5}\,dh}_{\text{differentiate } h^{-4}}\]
Dividing both sides by \(y = f^2 g^3 h^{-4}\):
\[\frac{dy}{y} = \frac{2\,df}{f} + \frac{3\,dg}{g} - \frac{4\,dh}{h}\]
Each term is just the exponent times the relative change of that factor. This is called logarithmic differentiation and makes complicated products much easier!
Composite Functions & the Chain Rule
A composite function is a function inside another function. For example:
\[y = (3x^2 + 1)^5\]
Here the “inner” function is \(u = 3x^2 + 1\) and the “outer” function is \(y = u^5\).
The Chain Rule
To find \(dy\), work from the outside in:
\[dy = 5u^4 \cdot du = 5(3x^2 + 1)^4 \cdot 6x\,dx\]
In fraction form:
To differentiate a function inside another function, multiply the derivative of the outer function (evaluated at the inner function) by the derivative of the inner function. It’s like a chain of gears — each rate multiplies the next.
\[\boxed{\frac{dy}{dx} = \frac{dy}{du} \cdot \frac{du}{dx}}\]
Think of it like a chain of gears: a tiny turn in \(x\) causes a turn in \(u\), which causes a turn in \(y\). The total effect multiplies.
Explore the chain rule — see how the inner function shapes the derivative:
Derivative of an Inverse Function
If \(y = f(x)\), the inverse function gives you \(x = f^{-1}(y)\) — it “undoes” \(f\).
The derivative of the inverse has a beautifully simple formula:
The slope of an inverse function is just the reciprocal of the original function’s slope. If the original goes up steeply, the inverse goes up gently, and vice versa.
\[\boxed{\frac{dx}{dy} = \frac{1}{\dfrac{dy}{dx}}}\]
This makes geometric sense: if \(y\) changes 3 times as fast as \(x\), then \(x\) changes \(\frac{1}{3}\) as fast as \(y\).
Example
If \(y = x^3\), then \(\frac{dy}{dx} = 3x^2\), so:
\[\frac{dx}{dy} = \frac{1}{3x^2}\]
Since \(x = y^{1/3}\), we can write \(x^2 = y^{2/3}\), giving \(\frac{dx}{dy} = \frac{1}{3y^{2/3}}\).
This matches what we’d get from differentiating \(x = y^{1/3}\) directly!
Monotonicity and Invertibility
A function is invertible only if it is monotonic — meaning it is always increasing or always decreasing. If it turns around (has a local max or min), then two different \(x\)-values give the same \(y\), and you can’t reverse the process uniquely.
How to Check: The Discriminant Test
For a polynomial like \(y = ax^3 + bx^2 + cx + d\), the derivative is:
\[\frac{dy}{dx} = 3ax^2 + 2bx + c\]
This is a quadratic. The function is monotonic if this quadratic never changes sign, which happens when its discriminant is negative:
\[(2b)^2 - 4(3a)(c) < 0 \quad \Longrightarrow \quad 4b^2 - 12ac < 0\]
No real roots means the derivative never crosses zero, so the function never turns around!
Explore — a cubic with adjustable coefficients. When is it invertible?
In the graph above, try setting \(a = 1\), \(b = 3\), \(c = 1\). You should see the derivative (dashed red curve) dip below zero — meaning the cubic has a local max and min, and is not invertible!
Now try \(a = 1\), \(b = 0\), \(c = 1\). The derivative stays positive everywhere — the cubic is strictly increasing and is invertible.
Cheat Sheet
| Rule | Formula |
|---|---|
| Product rule | \(d(fg) = g\,df + f\,dg\) |
| Quotient rule | \(d\!\left(\frac{f}{g}\right) = \frac{g\,df - f\,dg}{g^2}\) |
| Extended quotient | \(d\!\left(\frac{f}{g^2}\right) = \frac{g\,df - 2f\,dg}{g^3}\) |
| Chain rule | \(\frac{dy}{dx} = \frac{dy}{du} \cdot \frac{du}{dx}\) |
| Inverse function | \(\frac{dx}{dy} = \frac{1}{dy/dx}\) |
Generalized Product Rule
To differentiate \(f_1^{n_1} \cdot f_2^{n_2} \cdots f_k^{n_k}\): for each factor, differentiate it (bring down the exponent), hold the rest fixed, and add all pieces together.
Invertibility Check
A function is invertible when it is monotonic (always increasing or always decreasing). For a cubic \(ax^3 + bx^2 + cx + d\), check that the derivative \(3ax^2 + 2bx + c\) has a negative discriminant: \(4b^2 - 12ac < 0\).
Orders of Infinitesimals
Always keep the lowest-order terms. Drop any product of two or more infinitesimals (\(df \cdot dg\), \(dx^2\), etc.).