How Do the Product and Chain Rules Work?

TL;DR

The product rule adds the two first-order area changes: keep each factor once while differentiating the other. The chain rule multiplies the derivative of the outside function, evaluated at the unchanged inside function, by the derivative of that inside function. Both rules follow from tracking how tiny input nudges propagate through combined functions.

Transcript

In the last videos I talked about the derivatives of simple functions, and the goal was to have a clear picture or intuition to hold in your mind that actually explains where these formulas come from. But most of the functions you deal with in modeling the world involve mixing, combining, or tweaking these simple functions in some other way, so our... Read More

Key Insights

• Most complicated functions are layered from addition, multiplication, and composition. Once the derivative rules for these three combinations are understood, even a monstrous expression can be differentiated by working through its structure one layer at a time.
• The sum rule states that the derivative of a sum is the sum of the derivatives. A tiny input nudge changes each component separately, and adding those component changes gives the total change in the combined function.
• The derivative of sine of x plus x squared is cosine of x plus 2x. The sine term changes by approximately cosine of x times dx, while the squared term changes by approximately 2x times dx.
• The product rule comes from interpreting multiplication as the area of an adjustable rectangle. When both side lengths change slightly, the important area increase consists of two thin rectangles, each pairing one original side with the change in the other.
• The tiny corner area in the product-rule diagram becomes negligible as dx approaches zero. That corner depends on the product of two tiny changes, so it does not contribute to the first-order ratio used for the derivative.
• The product rule for g times h is g times the derivative of h plus h times the derivative of g. The mnemonic "left d right, right d left" records the two thin-area contributions in the rectangle model.
• The chain rule describes how a tiny change propagates through a composition. The inside function first converts dx into an intermediate change, and the outside function then converts that intermediate change into the final output change.
• The derivative of g composed with h is the derivative of g evaluated at h, multiplied by the derivative of h. The outside derivative keeps the unaltered inside function as its input because the outside function receives the intermediate value h.

Questions & Answers

Q: What are the basic ways functions can be combined?

Functions can be combined in three basic ways: addition, multiplication, and composition, which means placing one function inside another. Subtraction does not require a separate derivative rule because it can be written as multiplication by negative one followed by addition. Division can likewise be represented by composing with the function one over x and then multiplying.

Q: Why is the derivative of a sum the sum of derivatives?

A tiny input change causes a separate tiny output change in each function being added. Because the combined function is defined by adding their values, its total output change is also the sum of those individual changes. Dividing that total change by the original input change gives the sum of the two derivative ratios, which produces the sum rule.

Q: How is the product rule visualized with a rectangle?

A product of two functions can be treated as the area of a rectangle whose side lengths depend on x. A tiny change in x slightly changes both sides, creating a thin rectangle along the bottom, another along the side, and a tiny corner. The two thin rectangles provide the first-order change used in the product rule.

Q: What is the product rule for two functions?

For two functions g and h, the derivative of their product is g multiplied by the derivative of h, plus h multiplied by the derivative of g. Each term corresponds to one thin rectangle in the adjustable-area model. One uses the original width and changed height, while the other uses the original height and changed width.

Q: Why can the corner term be ignored in the product rule?

The corner has an area formed by multiplying two tiny side changes. Because both changes are proportional to a tiny input nudge, their product is proportional to dx squared. As dx approaches zero, this second-order contribution becomes negligible compared with the two thin rectangles, whose areas are proportional to dx and determine the derivative.

Q: How does the chain rule work for composed functions?

The chain rule tracks a small input change through two successive functions. The inside function converts dx into an intermediate change dh. The outside function then converts dh into the final output change. Since each conversion has its own proportionality factor, the overall derivative is found by multiplying the outside derivative by the inside derivative.

Q: What is the derivative of sine of x squared?

The derivative of sine of x squared is cosine of x squared multiplied by 2x. First, the outside sine function contributes cosine evaluated at the unchanged inside value, x squared. Then the inside function x squared contributes its derivative, 2x. Multiplying these factors gives the proportionality between the final output change and dx.

Q: Why is the outside derivative evaluated at the inside function?

The outside function does not receive x directly. It receives the value produced by the inside function, such as h equal to x squared. Therefore, when the outside derivative measures sensitivity at the current intermediate value, it must be evaluated at h. Replacing h with the original expression keeps the inside function unaltered within the outside derivative.

Summary & Key Takeaways

• Complicated functions can be understood as layers built from three basic combinations: addition, multiplication, and composition. Subtraction can be represented using multiplication by negative one and addition, while division can be represented through composition with one over x and multiplication. Differentiation can therefore proceed by peeling apart these layers step by step.

• The sum rule follows because a tiny change in a sum equals the sum of the tiny changes in its parts. For sine of x plus x squared, the total change is approximately cosine of x times dx plus 2x times dx, producing the derivative cosine of x plus 2x.

• The product rule is visualized through the changing area of a rectangle, while the chain rule is visualized with linked number lines. The area model produces two meaningful first-order rectangles, and the composition model shows successive changes multiplying as a nudge passes through the inside function and then the outside function.