Speed is a derivative in disguise
Your car speedometer never shows an average over the last hour; it shows how fast your position is changing right now. That instantaneous rate of change is the derivative. In geometry it is the slope of the tangent line; in physics it is velocity, acceleration, and much more.
The definition of the derivative
Formula: f'(x) = lim[h->0] [f(x+h) - f(x)] / h
The derivative measures how f changes as x changes: take the rise over a tiny run h, then let h shrink to zero. In practice you use shortcut rules: the power rule turns x^n into n x x^(n-1), constants differentiate to zero, and the chain rule handles nested functions.
Using the calculator
- Enter the function, for example x^2 or 3x^3 - 2x.
- Choose which variable to differentiate with respect to.
- Read the derivative, and evaluate it at a point if you need the slope there.
A worked example, step by step
Differentiate f(x) = x^2 and evaluate at x = 3.
- Step 1: apply the limit definition at x = 3: [(3+h)^2 - 9] / h.
- Step 2: expand: (9 + 6h + h^2 - 9) / h = (6h + h^2) / h.
- Step 3: divide through: 6 + h.
- Step 4: let h approach 0: the result is 6.
The derivative is 2x, and at x = 3 the slope is 6. The power rule agrees: 2 x 3 = 6.
Where derivatives matter
Engineers optimize designs with them, economists find marginal cost with them, and every tangent line starts here. The limit calculator covers the foundation underneath, the partial derivative calculator extends the idea to several variables, and the standard form calculator tidies up the tangent lines you find.
Mistakes that give the wrong answer
- Forgetting to lower the exponent. The derivative of x^3 is 3x^2, not 3x^3; each term loses one power.
- Skipping the chain rule. For something like (3x + 1)^2, you must multiply by the derivative of the inside, 3.
- Differentiating constants. A lone number has no change in it, so its derivative is zero, not one.
The second derivative: acceleration
Differentiate twice and you get the rate of change of the rate of change. If f(x) = x^3 describes position, then f'(x) = 3x^2 is velocity and f''(x) = 6x is acceleration. In graphing, the second derivative reveals concavity: positive means the curve cups upward like a bowl, negative means it arches downward. Optimization leans on this too. At a critical point where f'(x) = 0, a positive second derivative marks a minimum and a negative one marks a maximum.
What this assumes
Not every function has a derivative everywhere: sharp corners, jumps, and vertical tangents are points where the derivative does not exist.