1. Multiplying derivatives in a product
The most tempting wrong rule in calculus: differentiating each factor and multiplying the results. The product rule says (f·g)′ = f′g + fg′ — each factor takes its turn being differentiated while the other stays put, and the results are added.
The habit: any time you see two variable expressions multiplied, write the product rule template out before filling anything in.
2. Dropping the chain rule factor
Whenever the inside of a function is anything other than a bare x, its derivative must be multiplied on. Even a simple 2x inside counts.
The habit: after differentiating any function, ask “was the inside just x?” If not, the chain rule owes you a factor. Our chain rule guide covers this in depth.
3. Using the power rule on exponentials
The power rule applies when the base is the variable (xⁿ). When the variable is in the exponent (2ˣ, eˣ), it is an exponential function with a completely different rule: d/dx[aˣ] = aˣ·ln(a).
The habit: before reaching for the power rule, check where the variable lives. Base → power rule. Exponent → exponential rule.
4. Sign errors with sine and cosine
Sine differentiates to cosine with no sign change; cosine differentiates to negative sine. Half of all trig derivative errors are this one sign.
The habit: memorize the cycle sin → cos → −sin → −cos → sin. Differentiating moves you one step forward in the cycle, and the minus signs live on the second half.
5. Confusing ln(x) and 1/x derivatives
These two get tangled because each involves the other: the derivative of ln(x) is 1/x, but the derivative of 1/x is −1/x² (power rule with n = −1).
The habit: rewrite 1/x as x⁻¹ before differentiating. The power rule then does the work and the sign comes out automatically.
6. Differentiating constants as if they were variables
Constants — including symbolic ones like π and e — have derivative zero, no matter how exotic they look. π³ is just a number.
The habit: before differentiating any term, ask whether it actually contains the variable. No variable → derivative is zero, full stop.
7. Flipping the quotient rule numerator
The quotient rule’s numerator is f′g − fg′ — derivative of the top first. Because the terms are subtracted, swapping them silently negates your whole answer.
The habit: say it as you write it: “low d-high minus high d-low, over low squared.” Corny, but it has survived generations of calculus classes because it works. See our product & quotient rules guide for the full treatment.
How to check your own work
Beyond knowing the traps, two checking strategies catch nearly everything:
- Compare against a step-by-step solution. Don’t just compare final answers — find the first line where your work diverges. That line is the rule you need to review.
- Sanity-check numerically. Pick an easy value like
x = 1, estimate the slope of the original function there, and see whether your derivative gives roughly that number. A sign error or dropped factor shows up immediately.
The derivative calculator names every rule as it applies it, which makes the first strategy fast: paste in your expression, expand the steps, and find the exact point of disagreement.