What a Derivative Actually Measures
A derivative is the instantaneous rate of change of a function. If f(x) is a position, f′(x) is speed at that exact moment. If f(x) is a cost, f′(x) is the extra cost of one more unit. Geometrically it is the slope of the tangent line, the straight line that just touches the curve at one point.
The formal definition is a limit: f′(x) = limh→0 [f(x+h) − f(x)] ÷ h. You almost never use it by hand. Newton and Leibniz built the rules below in the 1660s and 1670s precisely so that nobody would have to.
f(x) = x², find f′(x)
- Write the difference quotient: [(x+h)² − x²] ÷ h.
- Expand: (x² + 2xh + h² − x²) ÷ h = (2xh + h²) ÷ h.
- Cancel h: 2x + h.
- Let h → 0: the limit is 2x.
Answer: f′(x) = 2x
The Rules You Need, in One Table
| Rule | Formula | Example |
|---|---|---|
| Constant | d/dx [c] = 0 | d/dx [7] = 0 |
| Power | d/dx [xⁿ] = n·xⁿ⁻¹ | d/dx [x⁵] = 5x⁴ |
| Sum | (f + g)′ = f′ + g′ | (x³ + 4x)′ = 3x² + 4 |
| Product | (fg)′ = f′g + fg′ | (x²eˣ)′ = (x² + 2x)eˣ |
| Quotient | (f/g)′ = (f′g − fg′) ÷ g² | (x/(x+1))′ = 1/(x+1)² |
| Chain | [f(g(x))]′ = f′(g(x))·g′(x) | (sin 3x²)′ = 6x cos 3x² |
| Exponential | (aˣ)′ = aˣ ln a | (2ˣ)′ = 2ˣ ln 2 |
The chain rule is the one that decides whether you pass. Think of a function as layers: peel the outside layer, differentiate it while leaving the inside alone, then multiply by the derivative of the inside. Every “my answer is off by a factor” error comes from forgetting that last multiplication.
Worked Examples With Real Numbers
Differentiate f(x) = (x² + 1) ÷ (x − 1)
- Name the parts: top u = x² + 1, bottom v = x − 1, so u′ = 2x and v′ = 1.
- Apply (u′v − uv′) ÷ v²: [2x(x − 1) − (x² + 1)(1)] ÷ (x − 1)².
- Expand the top: 2x² − 2x − x² − 1 = x² − 2x − 1.
Answer: f′(x) = (x² − 2x − 1) ÷ (x − 1)²
Differentiate y = x²·sin(3x)
- Product rule: y′ = 2x·sin 3x + x²·(sin 3x)′.
- Chain rule on the second factor: (sin 3x)′ = 3 cos 3x.
Answer: y′ = 2x sin 3x + 3x² cos 3x
Find the tangent to f(x) = x³ at x = 2
- f(2) = 8, so the point is (2, 8).
- f′(x) = 3x², so the slope is f′(2) = 12.
- Point-slope form: y − 8 = 12(x − 2).
Answer: y = 12x − 16
Using Derivatives: Speed, Peaks and Optimisation
A ball thrown straight up at 20 m/s has height s(t) = 20t − 4.9t² metres. The derivative s′(t) = 20 − 9.8t is its velocity. The ball stops rising when velocity is zero, at t = 20 ÷ 9.8 ≈ 2.04 s, and the height then is s ≈ 20.41 m. That is the whole idea of optimisation: set the derivative to zero, then check what kind of point you found.
| Test | What to do | Meaning |
|---|---|---|
| First derivative | Solve f′(x) = 0 | Candidates for a peak, dip or flat spot |
| Sign of f′ | Check on each side | + then − is a maximum, − then + is a minimum |
| Second derivative | Evaluate f″ at the point | f″ < 0 maximum, f″ > 0 minimum, f″ = 0 inconclusive |
The “Critical points” mode does all three steps and lists intervals where the function is increasing or decreasing.
What We Think Is Overrated, Underrated and Wrong
Derivatives People Search For
| Search | Answer | Method |
|---|---|---|
| d/dx xˣ | xˣ(ln x + 1) | logarithmic differentiation |
| d/dx √x | 1 ÷ (2√x) | power rule, exponent ½ |
| d/dx ln(x²) | 2 ÷ x | chain rule (or 2 ln x) |
| d/dx e²ˣ | 2e²ˣ | chain rule |
| d/dx aˣ | aˣ ln a | exponential rule |
| d/dx 1/x | −1 ÷ x² | power rule, exponent −1 |
| d/dx sin²x | 2 sin x cos x | chain rule |
| d/dx |x| | x ÷ |x| for x ≠ 0 | piecewise |
At x = 2, the derivative of xˣ is 2²(ln 2 + 1) = 6.7726. You cannot get that with the power rule, because the exponent also contains x.
Checking a Derivative Numerically
The symmetric difference [f(x + h) − f(x − h)] ÷ 2h approaches f′(x) as h shrinks. For f(x) = x² at x = 3 with h = 0.001: (9.006001 − 8.994001) ÷ 0.002 = 6, matching 2x = 6. If your symbolic answer disagrees with this number, one of them has a mistake.
Our Take: What Is Overrated, Underrated and Plain Wrong
Overrated: the limit definition drill. It builds understanding once, then the rules do the work.
Underrated: the numeric check above. It is a 10-second test that catches wrong chain-rule factors.
Plain wrong: the claim that d/dx eˣ = x·eˣ⁻¹. That applies the power rule to a variable exponent. The derivative of eˣ is eˣ.
Derivative Mistakes That Cost Marks
- Forgetting the inner derivative in the chain rule: (sin 3x)′ is 3 cos 3x, not cos 3x.
- Differentiating aˣ like a power: (2ˣ)′ is 2ˣ ln 2, not x·2ˣ⁻¹.
- Dropping a minus sign in the quotient rule numerator. It is u′v − uv′, in that order.
- Treating e as a variable: (eˣ)′ = eˣ, and the derivative of the constant e2 is 0.
- Mixing up f′(a), the slope at one point, with f′(x), the whole function.
Why Students Get Stuck on Derivatives, and What a Good Calculator Should Do
Searching for help with derivatives usually starts late at night with one problem that will not work out. The rules are known, but choosing between product, quotient and chain rule is hard. A good calculator fixes that by showing each step, checking its own answer and explaining the idea in plain words. This section covers what students struggle with, what they want, and how this page responds.
In short: students need to recognise which rule applies and apply it without dropping terms, not just see a final number, and a calculator earns trust only when its steps can be checked.
Understand the pain points
What students get stuck on
Most mistakes here are not about arithmetic. They come from one rule applied the wrong way round, which is why seeing every step matters.
- Using the product rule as if it were (fg)′ = f′g′Find the derivative of x^3 * sin(x)↗
- Dropping the inner derivative in the chain ruleFind the derivative of (x^2 + 1)/(x - 3)↗
- Wrong sign when differentiating cos(x) or a quotientFind the derivative of sin(x^2)↗
- Not knowing when to simplify before differentiatingFind the derivative of e^(2x) * ln(x)↗
Understand the frustrations and desires
What they feel, and what they want
Each frustration points to a wish: a clearer step, a built-in check, an explanation in plain words.
- Wants the rule named at every step
- Wants a tangent line or slope at a point
- Wants a quick numerical check
Build the page around their intent
Each intent has a place on this page
The page follows intent, not features, so each need has a clear starting point.
- I need an answer now: type the problem into the calculator at the top of this page.Find the derivative of x^3 * sin(x)↗
- I need to understand: the sections below explain the idea in plain words and list the common mistakes.
- I need to check my work: every step is shown, so the first line that differs from yours is the one to study.
- I need practice: open the solver, tap + and choose Create Practice Test.
- I need it explained aloud: in the solver, tap + and choose Create Video.