Derivative Calculator
Differentiate a function symbolically, see the derivative and second derivative, and get the slope and tangent line at any point.
About the Derivative Calculator
The derivative f′(x) gives the slope of f at every point: how fast it is changing. The calculator applies the standard rules (power, product, quotient and chain) and then simplifies the result.
Every result is checked by the same engine that draws the graph, so you can see f′ crossing zero where f has a peak or a dip.
The formula
f′(x) = lim h→0 (f(x + h) − f(x)) ÷ h
- Power rule
- d/dx xⁿ = n·xⁿ⁻¹
- Product rule
- (uv)′ = u′v + uv′
- Chain rule
- f(g(x))′ = f′(g(x))·g′(x)
- Quotient rule
- (u/v)′ = (u′v − uv′) ÷ v²
Common derivatives
| f(x) | f′(x) |
|---|---|
| xⁿ | n·xⁿ⁻¹ |
| eˣ | eˣ |
| ln x | 1/x |
| sin x | cos x |
| cos x | −sin x |
| tan x | sec² x |
| aˣ | aˣ·ln a |
Worked example
Example: d/dx [x^2 + 3x] = 2x + 3
- Inputs: Function: y = f(x); f(x) = x^2 + 3x; Evaluate at x = (optional): 2.
- Formula: f′(x) = lim h→0 (f(x + h) − f(x)) ÷ h
- Sum rule: d/dx [x^2 + 3x] = 2x + 3
- Power rule: d/dx [x^2] = 2x
- Constant multiple rule: d/dx [3x] = 3
- Result: d/dx [x^2 + 3x] = 2x + 3
- Second derivative f″(x): 2
- f′(2): 7
- f″(2): 2
Questions people ask
Why does the answer look different from my textbook?
There are many equivalent forms; for example 2sin(x)cos(x) equals sin(2x). Check by plugging in a value.
Does it handle trig in degrees?
Calculus uses radians: d/dx sin x = cos x only holds in radians.