Quick start
- Enter f(x), for example x^3 * sin(x) or 4x^2 - 5x + 1. Use ^ for powers, ln(x) for the natural log and e^x or exp(x) for the exponential.
- Set the order: 1 for f′, 2 for f″, and so on up to 8.
- To get a number, type a value in "Evaluate at".
The main rules
| Rule | Formula |
|---|---|
| Power | (xⁿ)′ = n·xⁿ⁻¹ |
| Product | (uv)′ = u′v + uv′ |
| Quotient | (u/v)′ = (u′v − uv′) ÷ v² |
| Chain | f(g(x))′ = f′(g(x)) · g′(x) |
Common results: (sin x)′ = cos x, (cos x)′ = −sin x, (eˣ)′ = eˣ and (ln x)′ = 1/x.
Example: f(x) = x² · eˣ
By the product rule, f′(x) = 2x·eˣ + x²·eˣ = eˣ(x² + 2x). At x = 1, the slope is e · 3 ≈ 8.155.
What the derivative tells you
The first derivative is the rate of change, or the slope of the tangent line. Where it is zero the curve is flat, which is where maxima and minima occur. The second derivative tells you whether the curve bends upward or downward.
Common questions
Does it work in degrees?
No. Calculus formulas for sin and cos only hold in radians, so the tool always uses radians.
What is the difference between ln and log?
ln is the natural log, base e. log means base 10.
Can I use t or another variable?
Yes. Type the letter in the Variable box and use it in the function.