Derivative Calculator — Differentiation Rules & Common Derivatives
Differentiate functions using power rule, chain rule, product and quotient rules. Includes a common derivatives reference table and worked examples for calculus students.
A derivative measures how fast a function is changing at any given point — the instantaneous rate of change. It's the slope of the tangent line to the curve. Once you know the differentiation rules, most derivatives follow patterns that become second nature. Verify your work with the CalcHub Derivative Calculator.
The Definition (What You're Actually Computing)
$$f'(x) = \lim_{h \to 0} \frac{f(x+h) - f(x)}{h}$$
In practice you almost never compute this limit directly — you use the rules below instead.
The Essential Differentiation Rules
Power Rule
$$\frac{d}{dx}[x^n] = nx^{n-1}$$d/dx[x³] = 3x², d/dx[x⁻²] = −2x⁻³, d/dx[√x] = d/dx[x^(1/2)] = (1/2)x^(−1/2)
Sum/Difference Rule
$$\frac{d}{dx}[f(x) \pm g(x)] = f'(x) \pm g'(x)$$Derivatives distribute across addition and subtraction.
Constant Rule
$$\frac{d}{dx}[c] = 0 \quad \text{and} \quad \frac{d}{dx}[cf(x)] = c \cdot f'(x)$$Product Rule
$$\frac{d}{dx}[f(x) \cdot g(x)] = f'(x)g(x) + f(x)g'(x)$$ Memory trick: "first times derivative of second, plus second times derivative of first" Example: d/dx[x²·sin x] = 2x·sin x + x²·cos xQuotient Rule
$$\frac{d}{dx}\left[\frac{f(x)}{g(x)}\right] = \frac{f'(x)g(x) - f(x)g'(x)}{[g(x)]^2}$$ Memory trick: "lo d-hi minus hi d-lo, over lo-lo" Example: d/dx[x/sin x] = (1·sin x − x·cos x) / sin²xChain Rule
$$\frac{d}{dx}[f(g(x))] = f'(g(x)) \cdot g'(x)$$Differentiate the outer function, keep the inner intact, then multiply by the derivative of the inner function.
Example: d/dx[sin(x²)] = cos(x²) · 2x = 2x cos(x²) Example: d/dx[(3x + 1)⁵] = 5(3x + 1)⁴ · 3 = 15(3x + 1)⁴Common Derivatives Reference Table
| Function f(x) | Derivative f'(x) |
|---|---|
| c (constant) | 0 |
| xⁿ | nxⁿ⁻¹ |
| eˣ | eˣ |
| aˣ | aˣ ln a |
| ln x | 1/x |
| log_a x | 1/(x ln a) |
| sin x | cos x |
| cos x | −sin x |
| tan x | sec²x |
| cot x | −csc²x |
| sec x | sec x tan x |
| csc x | −csc x cot x |
| arcsin x | 1/√(1−x²) |
| arccos x | −1/√(1−x²) |
| arctan x | 1/(1+x²) |
Worked Example: Full Problem
Differentiate f(x) = 3x⁴ − 2x³ + 5x − 7
f'(x) = 3·4x³ − 2·3x² + 5·1 − 0
f'(x) = 12x³ − 6x² + 5
To find the slope at x = 1: f'(1) = 12 − 6 + 5 = 11
Higher-Order Derivatives
- f'(x) = first derivative (velocity if f is position)
- f''(x) = second derivative (acceleration)
- f'''(x) = third derivative (jerk)
How do I know when to use the chain rule?
Use the chain rule whenever you have a function of a function — a "composition." If you can identify an inner function u(x) inside an outer function f(u), apply chain rule: d/dx[f(u)] = f'(u)·u'. Common triggers: anything raised to a power involving x, trig functions of expressions, e raised to an expression.
What's the practical meaning of the second derivative?
The second derivative tells you about concavity. f''(x) > 0 means the curve is concave up (bowl shape, like a minimum). f''(x) < 0 means concave down (hill shape, like a maximum). This is the second derivative test for identifying local minima and maxima.
Is implicit differentiation a separate rule?
Implicit differentiation is just the chain rule applied when y appears in the equation but can't be easily isolated. Differentiate both sides with respect to x, and whenever you differentiate a term involving y, multiply by dy/dx. Then solve for dy/dx algebraically.