Gilbert Strang’s Calculus: Highlights

Calculus

Derivatives

Integrals

Graphs

Author

Chao Ma

Published

February 1, 2026

“Calculus is about pairs of functions.” — Gilbert Strang

Reference: Highlights of Calculus (MIT OCW)

Derivatives: rates of change

Calculus connects a function to its rate of change. The derivative tells us how fast $y$ changes when $x$ changes.

Basic rules

Constant: $\dfrac{d}{dx}(u+c)=\dfrac{du}{dx}$
Sum: $\dfrac{d}{dx}(u+v)=\dfrac{du}{dx}+\dfrac{dv}{dx}$
Product: $\dfrac{d}{dx}(uv)=u\,\dfrac{dv}{dx}+v\,\dfrac{du}{dx}$

Three example functions

Linear: $y=2x$ (constant growth)
Quadratic: $y=x^2$ (growth increases with $x$)
Exponential: $y=2^x$ (growth increases explosively)

These have very different growth rates, even though they all increase.

Calculating change

There are two perspectives on change:

Limits: $\dfrac{\Delta y}{\Delta x}$, then take $\Delta x\to 0$
Rules: start from known derivatives and build new ones

For a linear combination, \[ \frac{d}{dx}(C_1y_1+C_2y_2)=C_1\frac{dy_1}{dx}+C_2\frac{dy_2}{dx}. \]

Example: if $y=x^2-1$, then $\dfrac{dy}{dx}=2x$.

Slope and the tangent line

The average slope between two points is \[ \frac{y_2-y_1}{x_2-x_1}=\frac{\Delta y}{\Delta x}. \] The instantaneous slope is the limit as $\Delta x\to 0$.

For $y=x^2$, \[ \frac{\Delta y}{\Delta x}= \frac{(x+\Delta x)^2-x^2}{\Delta x}=2x+\Delta x \quad\Rightarrow\quad \frac{dy}{dx}=2x. \]

This produces the linear approximation near $x_0$: \[ y\approx y_0+\frac{dy}{dx}(x_0)(x-x_0). \]

Second derivative

The second derivative is the slope of the slope. If $y=x^2$, then $\dfrac{dy}{dx}=2x$ and $\dfrac{d^2y}{dx^2}=2$. It tells us whether the function is speeding up or slowing down.

Power rule

For $y=x^n$, \[ \frac{dy}{dx}=n x^{n-1}. \]

Examples: - $y=x^2 \Rightarrow \dfrac{dy}{dx}=2x$ - $y=x^3 \Rightarrow \dfrac{dy}{dx}=3x^2$ - $y=x^4 \Rightarrow \dfrac{dy}{dx}=4x^3$

Sketch of proof (induction): Assume $\dfrac{d}{dx}x^n=nx^{n-1}$. Then \[ \frac{d}{dx}(x\cdot x^n)=1\cdot x^n + x\cdot (n x^{n-1})=(n+1)x^n. \]

Exponential function

The exponential function $y=e^x$ solves \[ \frac{dy}{dx}=y. \] It equals its own slope.

Series expansion: \[ e^1=e=1+1+\frac{1}{2}+\frac{1}{6}+\frac{1}{24}+\dots \]

Key rules: - $e^{x_1}e^{x_2}=e^{x_1+x_2}$ - $(e^x)^n=e^{xn}$

Maxima and minima

To find extrema of $y=f(x)$, solve \[ \frac{dy}{dx}=0 \] for critical points. The second derivative distinguishes max vs. min.

Integrals: accumulation

If the derivative is speed, the integral is distance. If the derivative is a rate, the integral is the total effect.

The graph already contains the integral: the area under the curve measures accumulation.

Graphs as reasoning tools

In Highlights of Calculus, graphs are not for decoration—they are reasoning tools. Slope reveals the first derivative, concavity reveals the second derivative, and extrema and inflection points become visible directly from shape.

Takeaway. Calculus links pairs: a function and its derivative, or a rate and its accumulation. These connections let us read change from shape and compute global behavior from local rules.

--- title: "Gilbert Strang's Calculus: Highlights" author: "Chao Ma" date: "2026-02-01" categories: [Calculus, Derivatives, Integrals, Graphs] --- > "Calculus is about pairs of functions." — Gilbert Strang Reference: [Highlights of Calculus (MIT OCW)](https://ocw.mit.edu/courses/res-18-001-calculus-fall-2023/mitres_18_001_f17_full_book.pdf#page=13.15) # Derivatives: rates of change Calculus connects a function to its **rate of change**. The derivative tells us how fast $y$ changes when $x$ changes. ## Basic rules - **Constant**: $\dfrac{d}{dx}(u+c)=\dfrac{du}{dx}$ - **Sum**: $\dfrac{d}{dx}(u+v)=\dfrac{du}{dx}+\dfrac{dv}{dx}$ - **Product**: $\dfrac{d}{dx}(uv)=u\,\dfrac{dv}{dx}+v\,\dfrac{du}{dx}$ ## Three example functions - Linear: $y=2x$ (constant growth) - Quadratic: $y=x^2$ (growth increases with $x$) - Exponential: $y=2^x$ (growth increases explosively) These have very different growth rates, even though they all increase. ![Different growth rates](highlights-of-calculus-1.png) # Calculating change There are two perspectives on change: - **Limits**: $\dfrac{\Delta y}{\Delta x}$, then take $\Delta x\to 0$ - **Rules**: start from known derivatives and build new ones For a linear combination, $$ \frac{d}{dx}(C_1y_1+C_2y_2)=C_1\frac{dy_1}{dx}+C_2\frac{dy_2}{dx}. $$ Example: if $y=x^2-1$, then $\dfrac{dy}{dx}=2x$. # Slope and the tangent line The **average slope** between two points is $$ \frac{y_2-y_1}{x_2-x_1}=\frac{\Delta y}{\Delta x}. $$ The **instantaneous slope** is the limit as $\Delta x\to 0$. For $y=x^2$, $$ \frac{\Delta y}{\Delta x}= \frac{(x+\Delta x)^2-x^2}{\Delta x}=2x+\Delta x \quad\Rightarrow\quad \frac{dy}{dx}=2x. $$ This produces the **linear approximation** near $x_0$: $$ y\approx y_0+\frac{dy}{dx}(x_0)(x-x_0). $$ ## Second derivative The **second derivative** is the slope of the slope. If $y=x^2$, then $\dfrac{dy}{dx}=2x$ and $\dfrac{d^2y}{dx^2}=2$. It tells us whether the function is speeding up or slowing down. # Power rule For $y=x^n$, $$ \frac{dy}{dx}=n x^{n-1}. $$ Examples: - $y=x^2 \Rightarrow \dfrac{dy}{dx}=2x$ - $y=x^3 \Rightarrow \dfrac{dy}{dx}=3x^2$ - $y=x^4 \Rightarrow \dfrac{dy}{dx}=4x^3$ **Sketch of proof (induction):** Assume $\dfrac{d}{dx}x^n=nx^{n-1}$. Then $$ \frac{d}{dx}(x\cdot x^n)=1\cdot x^n + x\cdot (n x^{n-1})=(n+1)x^n. $$ # Exponential function The exponential function $y=e^x$ solves $$ \frac{dy}{dx}=y. $$ It equals its own slope. Series expansion: $$ e^1=e=1+1+\frac{1}{2}+\frac{1}{6}+\frac{1}{24}+\dots $$ Key rules: - $e^{x_1}e^{x_2}=e^{x_1+x_2}$ - $(e^x)^n=e^{xn}$ # Maxima and minima To find extrema of $y=f(x)$, solve $$ \frac{dy}{dx}=0 $$ for **critical points**. The second derivative distinguishes max vs. min. # Integrals: accumulation If the derivative is speed, the integral is distance. If the derivative is a rate, the integral is the total effect. The graph already contains the integral: **the area under the curve measures accumulation.** # Graphs as reasoning tools In *Highlights of Calculus*, graphs are not for decoration—they are reasoning tools. Slope reveals the first derivative, concavity reveals the second derivative, and extrema and inflection points become visible directly from shape. ![Reading derivatives from graphs](highlights-of-calculus-2.png) --- **Takeaway.** Calculus links pairs: a function and its derivative, or a rate and its accumulation. These connections let us read change from shape and compute global behavior from local rules.