Derivative Formula

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

The Formula

f′(x)=lim⁡h→0f(x+h)−f(x)hf'(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)=t2s(t) = t^2, velocity v(t)=s′(t)=2tv(t) = s'(t) = 2t At t=3t = 3, velocity =6= 6.

Notation

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

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→0f(a+h)−f(a)hf'(a) = \lim_{h \to 0} \frac{f(a+h) - f(a)}{h}, provided the limit exists. Equivalently, f′(a)=lim⁡x→af(x)−f(a)x−af'(a) = \lim_{x \to a} \frac{f(x) - f(a)}{x - a}.

Worked Examples

Example 1

easy
Find the derivative of f(x)=3x2+5x−2f(x) = 3x^2 + 5x - 2.

Answer

f′(x)=6x+5f'(x) = 6x + 5

First step

1
Apply the power rule to each term: ddx[xn]=nxn−1\frac{d}{dx}[x^n] = nx^{n-1}.

Full solution

  1. 2
    For 3x23x^2: 3⋅2x2−1=6x3 \cdot 2x^{2-1} = 6x.
  2. 3
    For 5x5x: 5⋅1x1−1=55 \cdot 1x^{1-1} = 5.
  3. 4
    For −2-2: the derivative of a constant is 00.
  4. 5
    Combine: f′(x)=6x+5f'(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)=x3−4x2+7xf(x) = x^3 - 4x^2 + 7x and evaluate f′(2)f'(2).

Example 3

hard
Use the limit definition to find the derivative of f(x)=x2f(x) = x^2.

Common Mistakes

  • Forgetting the limit and just computing f(x+h)−f(x)h\frac{f(x+h)-f(x)}{h} with a fixed hh — the derivative requires h→0h\to 0, collapsing the secant into a tangent.
  • Confusing f′(x)f'(x) (a function of xx) with f′(a)f'(a) (a single number) — evaluate the derivative function at the point to get the slope there.
  • Treating the derivative as the function's value instead of its rate — f(2)f(2) is the height of the curve, f′(2)f'(2) is its steepness.

Common Mistakes Guide

If this formula feels simple in isolation but keeps breaking during real problems, review the most common errors before you practice again.

Chain Rule MistakesProduct Rule Mistakes

Why This Formula Matters

The derivative converts a static curve into rate information: velocity from position, marginal cost from cost, where a function rises or falls. Students who only ever compute average slopes over intervals miss the whole point — the derivative is what lets you talk about the rate at a single instant where the interval has shrunk to nothing. Recognizing it by "Am I asked for the rate of change at a single instant, found by letting the gap hh shrink to zero?" — rather than by familiar numbers — is what lets a student tell it apart from average rate of change and slope of a line and integral in a mixed problem set.

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

Why is the Derivative formula important in Math?

The derivative converts a static curve into rate information: velocity from position, marginal cost from cost, where a function rises or falls. Students who only ever compute average slopes over intervals miss the whole point — the derivative is what lets you talk about the rate at a single instant where the interval has shrunk to nothing. Recognizing it by "Am I asked for the rate of change at a single instant, found by letting the gap hh shrink to zero?" — rather than by familiar numbers — is what lets a student tell it apart from average rate of change and slope of a line and integral in a mixed problem set.

What do students get wrong about Derivative?

The procedure for derivative is the easy part; the trap is forgetting the limit and just computing f(x+h)−f(x)h\frac{f(x+h)-f(x)}{h} with a fixed hh. Asking "Am I asked for the rate of change at a single instant, found by letting the gap hh shrink to zero?" first is what keeps a correct-looking calculation from being attached to the wrong concept.

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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