2.7 Derivatives of cos x, sin x, e^x, and ln x

Trig derivatives

For sin x, the derivative is cos x. For cos x, the derivative is -sin x. These come from limit definitions using trig identities, but memorize them directly for use in the chain and product rules. Notice the sign flip only appears with cosine.

Exponential derivative

The function e^x is unique because its derivative equals itself: d/dx[e^x] = e^x. This is essentially the *definition* of e as the base whose exponential growth rate matches its own value at every point — a key property used constantly in BC problems involving growth/decay.

Natural log derivative

The derivative of ln x is 1/x, valid for x > 0. This result follows from treating ln x as the inverse of e^x and applying implicit differentiation. It's frequently combined with the chain rule when the argument is more complex, like ln(3x² + 1).

Key terms

d/dx[sin x]

cos x

d/dx[cos x]

-sin x

d/dx[e^x]

e^x (its own derivative)

d/dx[ln x]

1/x, for x > 0

Why is e^x special?

It's the only function (up to scalar multiples) whose derivative equals itself, making it foundational for growth/decay models.

Try an AP-style question

d/dx (3 sin x − 2 cos x) = ?

1. 3 cos x + 2 sin x
2. −3 cos x + 2 sin x
3. 3 cos x + 2 cos x
4. 3 cos x − 2 sin x