Polynomial Derivative Calculator
Enter a polynomial like 3x^3 − 2x + 5 and get its derivative, with the power rule applied to each term, step by step.
How to use this calculator
Type your polynomial using ^ for powers (3x^3 means 3x³) and press Calculate. The calculator differentiates term by term with the power rule and shows each term’s transformation, so you can follow, or check, the working.
Scope: it handles polynomials, trigonometric, exponential, and logarithmic functions, with rule-named steps.
How it works
The power rule says d/dx[xⁿ] = n·xⁿ⁻¹: multiply by the exponent, then reduce the exponent by one. Constants vanish (their rate of change is zero), and sums differentiate term by term. That is the entire machinery behind this calculator.
Worked example
Example: f(x) = 3x³ − 2x + 5
Term 1: d/dx[3x³] = 3 × 3·x² = 9x².
Term 2: d/dx[−2x] = −2 × 1·x⁰ = −2.
Term 3: d/dx[5] = 0.
Answer: f′(x) = 9x² − 2.
Common mistakes
- Forgetting to reduce the exponent: d/dx[x³] is 3x², not 3x³, multiply and subtract one.
- Differentiating the constant: the derivative of a lone number is 0, not the number itself.
- Power rule on the wrong shape: it applies to xⁿ with constant n. For 2ˣ or xˣ you need exponential/logarithmic rules instead.
When to use it
Derivatives are instantaneous rates of change: velocity from position, marginal cost from total cost, the slope of a curve at a point, and the critical points behind every optimization problem (“maximize area”, “minimize material”).
Ready for harder shapes? See the derivative of xˣ worked by logarithmic differentiation, and a full quotient rule example for (x²+1)/(x−1).
Frequently asked questions
d/dx[xⁿ] = n·xⁿ⁻¹. Bring the exponent down as a multiplier, then subtract one from the exponent. Example: d/dx[x⁴] = 4x³.
Zero. Constants do not change, so their rate of change is 0, which is why the +5 disappears when differentiating 3x³ − 2x + 5.
Yes, trig, exp, and log functions are supported with rule-named steps, including the chain rule for composites like sin(x²).
The slope of the tangent line at each point. Where f′(x) = 0, the curve flattens, marking maxima, minima, or inflection points.
Derivative calculator with steps
Our derivative calculator shows every step using the power rule, applied term by term with the coefficient and exponent changes explained.
More derivative questions
Implicit differentiation finds dy/dx when y is mixed into the equation rather than isolated, like x^2 + y^2 = 25. Differentiate every term with respect to x, attach dy/dx to each y term via the chain rule, then solve for dy/dx.