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Intuition behind calculus rules

5 min read

From: 3blue1brown

Understanding calculus rules goes beyond mere memorization; the goal is to develop a clear picture or intuition that explains their origins [00:00:18]. This article explores the intuitive basis for the fundamental rules of differentiation: the sum rule, product rule, and chain rule [00:01:42].

Basic Function Combinations

Most functions encountered in modeling involve mixing, combining, or tweaking simpler functions [00:00:26]. These combinations primarily boil down to three basic operations [00:00:49]:

  1. Addition: Adding functions together [00:00:54].
  2. Multiplication: Multiplying functions [00:00:54].
  3. Composition: Plugging one function inside another [00:00:56].

Other operations like subtraction and division can be reduced to these three basic forms [00:01:00]:

  • Subtracting functions is equivalent to multiplying the second by -1 and adding them [00:01:00].
  • Dividing functions is the same as composing one function with 1/x and then multiplying the two functions [00:01:08].

Understanding how derivatives interact with these three combination types allows one to systematically approach the derivative of any complex expression [00:01:27].

The Sum Rule

The sum rule states that the derivative of a sum of two functions is the sum of their derivatives [00:01:54].

Intuition

Consider a function f(x) = sin(x) + x^2 [00:02:16]. For any input x, the value of f(x) is the sum of sin(x) and x^2 at that point [00:02:22]. Visually, if sin(x) and x^2 represent heights, their sum f(x) is obtained by stacking these heights [00:02:44].

When the input x is nudged slightly to x + dx, the total change in f(x) (denoted df) is simply the sum of the change in sin(x) (d sin(x)) and the change in x^2 (d x^2) [00:03:04].

  • The change d sin(x) is approximately cos(x) * dx (since the derivative of sine is cosine) [00:03:22].
  • The change d x^2 is approximately 2x * dx (since the derivative of x^2 is 2x) [00:03:43].

Therefore, df = (cos(x) * dx) + (2x * dx) [00:03:55]. Dividing by dx gives the derivative df/dx = cos(x) + 2x, which is the sum of the individual derivatives [00:03:55].

The Product Rule

The product rule is less straightforward than the sum rule [00:04:11].

Intuition

To understand the derivative of a product, such as f(x) = sin(x) * x^2, it’s helpful to visualize it as an area [00:04:29]. Imagine a rectangle where one side length is sin(x) and the other is x^2 [00:04:36]. The area of this rectangle is f(x) [00:05:20].

When the input x changes by a tiny dx, both sin(x) and x^2 change:

This change results in three small snippets of new area [00:05:50]:

  1. Bottom rectangle: Width sin(x) multiplied by the thin height d x^2 [00:05:53].
  2. Right rectangle: Height x^2 multiplied by the thin width d sin(x) [00:06:01].
  3. Corner square: Area is d sin(x) * d x^2. This term becomes negligible as dx approaches zero (it’s proportional to dx^2) [00:06:10].

So, the total change in area df is approximately sin(x) * d x^2 + x^2 * d sin(x) [00:07:02]. Substituting the derivatives:

Thus, df = sin(x) * (2x * dx) + x^2 * (cos(x) * dx) [00:07:02]. Dividing by dx yields the product rule: df/dx = sin(x) * (2x) + x^2 * (cos(x)) [00:07:09].

This pattern, often remembered as “left d right, right d left” [00:07:32], applies to any two functions g(x) and h(x): (gh)' = g'h + gh' [00:07:21]. Each term in the rule corresponds to the area of one of the thin rectangles [00:08:06].

For multiplication by a constant, like 2 * sin(x), the derivative is simply the constant multiplied by the derivative of the function (e.g., 2 * cos(x)) [00:08:20].

The Chain Rule

Function composition involves nesting one function inside another, such as f(x) = sin(x^2) [00:08:53].

Intuition

Visualize the process using three number lines [00:09:13]:

  1. Top line: Represents the input x [00:09:16].
  2. Middle line: Represents the inner function’s output, x^2 (let’s call this h) [00:09:18].
  3. Bottom line: Represents the final output, sin(x^2) (or sin(h)) [00:09:21].

A change in x by dx causes a change in h (i.e., x^2) by dh (or dx^2) [00:10:08]. This dh then causes a change in sin(h) by d sin(h) [00:10:42].

  • The change dh is approximately (derivative of x^2) * dx = 2x * dx [00:10:16].
  • The change d sin(h) is approximately (derivative of sin(h) with respect to h) * dh = cos(h) * dh [00:11:10].

Substituting h with x^2: d sin(x^2) is approximately cos(x^2) * dx^2 [00:11:19]. Further substituting dx^2 with 2x * dx: d sin(x^2) is approximately cos(x^2) * (2x * dx) [00:11:31].

To find the derivative df/dx, we divide d sin(x^2) by dx: df/dx = cos(x^2) * 2x [00:12:02].

This pattern is the chain rule: for a composition g(h(x)), its derivative is g'(h(x)) * h'(x) [00:12:29]. Expressed in Leibniz notation, dg/dx = (dg/dh) * (dh/dx) [00:12:52]. This form highlights the “cancellation” of dh, which is not just a notational trick but reflects how tiny changes propagate through the chain of functions [00:14:23].

Conclusion

The sum rule, product rule, and chain rule are the core tools for differentiating combined functions [00:14:36]. Understanding the intuitive origins of these rules—whether through adding changes, visualizing areas, or tracing cascades of nudges—transforms them from memorized formulas into natural patterns [00:14:15]. While conceptual understanding is crucial, practical application and practice are essential for fluency in calculus computations [00:15:06].

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