Gilbert Strang’s Calculus: Big Picture on Derivatives

Calculus

Derivatives

Slopes

Limits

Author

Chao Ma

Published

February 8, 2026

Video: Big Picture: Derivatives (Gilbert Strang)

Why derivatives matter

Derivatives connect a function to its local behavior: how fast it changes and which way it is heading. This note is a compact map of that idea using the most important functions and the slope viewpoint.

The great functions

These are the core families you keep seeing in calculus:

Power: \(y=x^n\)
- \(\dfrac{dy}{dx}=n x^{n-1}\)
Trigonometric:
- \(y=\sin x \Rightarrow \dfrac{dy}{dx}=\cos x\)
- \(y=\cos x \Rightarrow \dfrac{dy}{dx}=-\sin x\)
Exponential:
- \(y=e^x \Rightarrow \dfrac{dy}{dx}=e^x\)

Slope: average vs. instantaneous

Slope is the change in \(y\) over the change in \(x\).

Average slope over a small step \(\Delta x\): \[ \text{Average slope} = \frac{f(x+\Delta x)-f(x)}{\Delta x} \]

Instantaneous slope is the limit as \(\Delta x \to 0\): \[ \frac{dy}{dx}=\lim_{\Delta x \to 0}\frac{f(x+\Delta x)-f(x)}{\Delta x} \]

At the bottom of a curve, the slope is zero.

Slope at the bottom

Example: \(f(x)=x^2\)

Compute the instantaneous slope: \[ \lim_{\Delta x \to 0}\frac{(x+\Delta x)^2-x^2}{\Delta x} =\lim_{\Delta x \to 0}\frac{2x\Delta x+\Delta x^2}{\Delta x} =2x \]

So \(f'(x)=2x\).

Slope of a parabola

Function and derivative

Derivatives are functions too. A curve and its slope curve tell a two-part story.

\(f(x)=x^2\) and \(f'(x)=2x\)

Parabola and its derivative

\(f(x)=\sin x\) and \(f'(x)=\cos x\)

Sine and cosine

Takeaway. The derivative is the bridge from shape to change. Once you know the core functions and the slope limit, you can read local behavior directly from a graph.

---
title: "Gilbert Strang's Calculus: Big Picture on Derivatives"
author: "Chao Ma"
date: "2026-02-08"
categories: [Calculus, Derivatives, Slopes, Limits]
---

Video: [Big Picture: Derivatives (Gilbert Strang)](https://www.youtube.com/watch?v=T_I-CUOc_bk)

{{< video https://www.youtube.com/watch?v=T_I-CUOc_bk >}}

# Why derivatives matter

Derivatives connect a function to its local behavior: **how fast it changes** and **which way it is heading**. This note is a compact map of that idea using the most important functions and the slope viewpoint.

# The great functions

These are the core families you keep seeing in calculus:

- **Power**: $y=x^n$
  - $\dfrac{dy}{dx}=n x^{n-1}$
- **Trigonometric**:
  - $y=\sin x \Rightarrow \dfrac{dy}{dx}=\cos x$
  - $y=\cos x \Rightarrow \dfrac{dy}{dx}=-\sin x$
- **Exponential**:
  - $y=e^x \Rightarrow \dfrac{dy}{dx}=e^x$

# Slope: average vs. instantaneous

Slope is the change in $y$ over the change in $x$.

**Average slope** over a small step $\Delta x$:
$$
\text{Average slope} = \frac{f(x+\Delta x)-f(x)}{\Delta x}
$$

**Instantaneous slope** is the limit as $\Delta x \to 0$:
$$
\frac{dy}{dx}=\lim_{\Delta x \to 0}\frac{f(x+\Delta x)-f(x)}{\Delta x}
$$

At the bottom of a curve, the slope is zero.

![Slope at the bottom](big-picture-derivatives-1.png)

## Example: $f(x)=x^2$

Compute the instantaneous slope:
$$
\lim_{\Delta x \to 0}\frac{(x+\Delta x)^2-x^2}{\Delta x}
=\lim_{\Delta x \to 0}\frac{2x\Delta x+\Delta x^2}{\Delta x}
=2x
$$

So $f'(x)=2x$.

![Slope of a parabola](big-picture-derivatives-2.png)

# Function and derivative

Derivatives are functions too. A curve and its slope curve tell a two-part story.

- $f(x)=x^2$ and $f'(x)=2x$

![Parabola and its derivative](big-picture-derivatives-3.png)

- $f(x)=\sin x$ and $f'(x)=\cos x$

![Sine and cosine](big-picture-derivatives-4.png)

---

**Takeaway.** The derivative is the bridge from shape to change. Once you know the core functions and the slope limit, you can read local behavior directly from a graph.