Derivative Formula

The Formula

f'(x) = \lim_{h \to 0} \frac{f(x+h) - f(x)}{h}

When to use: How fast is the output changing right now? The slope of the curve at each point.

Quick Example

If position s(t) = t^2, velocity v(t) = s'(t) = 2t At t = 3, velocity = 6.

Notation

f'(x), \frac{dy}{dx}, \frac{df}{dx}, or Df(x) all denote the derivative of f with respect to x.

What This Formula Means

The instantaneous rate of change of a function at a single point, defined as the limit of the slope of secant lines.

How fast is the output changing right now? The slope of the curve at each point.

Formal View

f'(a) = \lim_{h \to 0} \frac{f(a+h) - f(a)}{h}, provided the limit exists. Equivalently, f'(a) = \lim_{x \to a} \frac{f(x) - f(a)}{x - a}.

Worked Examples

Example 1

easy
Find the derivative of f(x) = 3x^2 + 5x - 2.

Solution

  1. 1
    Apply the power rule to each term: \frac{d}{dx}[x^n] = nx^{n-1}.
  2. 2
    For 3x^2: 3 \cdot 2x^{2-1} = 6x.
  3. 3
    For 5x: 5 \cdot 1x^{1-1} = 5.
  4. 4
    For -2: the derivative of a constant is 0.
  5. 5
    Combine: f'(x) = 6x + 5.

Answer

f'(x) = 6x + 5
The power rule is the most fundamental differentiation rule. Apply it term by term and remember that the derivative of a constant is zero.

Example 2

medium
Find the derivative of f(x) = x^3 - 4x^2 + 7x and evaluate f'(2).

Example 3

hard
Use the limit definition to find the derivative of f(x) = x^2.

Common Mistakes

  • Confusing the power rule exponent: the derivative of x^n is nx^{n-1}, not nx^n — the exponent decreases by 1.
  • Treating the derivative of a product as the product of derivatives: \frac{d}{dx}[f(x)g(x)] \neq f'(x)g'(x) — you must use the product rule.
  • Forgetting that the derivative of a constant is 0, not 1 — constants have no rate of change.

Why This Formula Matters

Derivatives describe motion, enable optimization, and quantify how any quantity changes with respect to another.

Frequently Asked Questions

What is the Derivative formula?

The instantaneous rate of change of a function at a single point, defined as the limit of the slope of secant lines.

How do you use the Derivative formula?

How fast is the output changing right now? The slope of the curve at each point.

What do the symbols mean in the Derivative formula?

f'(x), \frac{dy}{dx}, \frac{df}{dx}, or Df(x) all denote the derivative of f with respect to x.

Why is the Derivative formula important in Math?

Derivatives describe motion, enable optimization, and quantify how any quantity changes with respect to another.

What do students get wrong about Derivative?

Derivative of position is velocity. Derivative of velocity is acceleration.

What should I learn before the Derivative formula?

Before studying the Derivative formula, you should understand: limit, slope.

Want the Full Guide?

This formula is covered in depth in our complete guide:

Derivatives Explained: Rules, Interpretation, and Applications →
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