Sines and Exponentials

Where e comes from

An interactive lesson with 7 steps, about 6 minutes: you predict, drag and solve, with instant feedback on every answer. It runs in your browser and needs JavaScript.

What you’ll learn

• $\frac{d}{dx}\sin x = \cos x$ and $\frac{d}{dx}\cos x = -\sin x$ (in radians).
• $e \approx 2.718$ is the base whose exponential has slope 1 at 0.
• $\frac{d}{dx}e^x = e^x$: the slope equals the height.

Steps

1. Slope of sine at 0
2. Tracing the slope
3. A slope on the sine wave
4. The perfect base
5. Its own derivative
6. Steepness of e^x
7. Slope of cosine