Proof
- |x| = x for x > 0 and −x for x < 0, so the derivative is 1 or −1.
- Both are x/|x|. At x = 0 the left and right derivatives (−1 and 1) differ, so |x| isn't differentiable there.
Standard derivatives
| Function | Derivative |
|---|---|
| d/dx(sin x) | cos x |
| d/dx(cos x) | −sin x |
| d/dx(tan x) | sec²x |
| d/dx(cot x) | −cosec²x |
| d/dx(sec x) | sec x tan x |
| d/dx(cosec x) | −cosec x cot x |
| d/dx(eˣ) | eˣ |
| d/dx(aˣ) | aˣ log a |
| d/dx(log x) | 1/x |
| d/dx(xⁿ) | n xⁿ⁻¹ |
| d/dx(√x) | 1/(2√x) |
| d/dx(1/x) | −1/x² |
| d/dx(sin⁻¹x) | 1/√(1 − x²) |
| d/dx(cos⁻¹x) | −1/√(1 − x²) |
| d/dx(tan⁻¹x) | 1/(1 + x²) |
| d/dx(cot⁻¹x) | −1/(1 + x²) |
| d/dx(sec⁻¹x) | 1/(|x|√(x² − 1)) |
| d/dx(xˣ) | xˣ(1 + log x) |
| d/dx(logₐx) | 1/(x log a) |
| d/dx(sin²x) | sin 2x |
| d/dx(log(sin x)) | cot x |
| d/dx(|x|) | x/|x| (x ≠ 0) |
Questions students ask
What is the derivative of |x|?
d/dx(|x|) = x/|x| (x ≠ 0).
How do you differentiate |x|?
|x| = x for x > 0 and −x for x < 0, so the derivative is 1 or −1. Both are x/|x|. At x = 0 the left and right derivatives (−1 and 1) differ, so |x| isn't differentiable there.