← Unit 2: Differentiation — Definition & Fundamental Properties
Derivatives of Trig, Exponential, and Log Functions
A short list of derivative formulas for sin, cos, tan, eˣ, and ln(x) needs to be memorized cold — they show up constantly, often nested inside other rules.
The must-know derivatives
d/dx[sin x] = cos x, and d/dx[cos x] = -sin x (note the sign flip). d/dx[tan x] = sec²x.
d/dx[eˣ] = eˣ — the exponential function is its own derivative, which is what makes it so special. For a general base, d/dx[aˣ] = aˣ·ln(a).
d/dx[ln x] = 1/x (for x > 0). These combine with the product, quotient, and (soon) chain rules just like any other function.
Worked Example
Differentiate f(x) = 3cos(x) - 5eˣ + ln(x).
- d/dx[3cos x] = 3·(-sin x) = -3sin x
- d/dx[-5eˣ] = -5eˣ
- d/dx[ln x] = 1/x
Answer: f'(x) = -3sin(x) - 5eˣ + 1/x
Key terms (4)
- d/dx[sin x]
- cos x
- d/dx[cos x]
- -sin x
- d/dx[eˣ]
- eˣ
- d/dx[ln x]
- 1/x
Practice Quiz
Question 1 of 3Score so far: 0/0
What is d/dx[sin(x)] at x = 0?