Enter a polynomial like 3x^3 + 2x^2 - 5x + 7 to get its derivative with per-term power rule steps. Covers polynomials; for trig or exponential functions, see the learning resources below.
Use the format: ax^n + bx^m + ... + c. Coefficients and integer exponents only. Example: 5x^4 - 3x^2 + x - 2
This is the default expression in the calculator. Here is each step applied by hand, term by term.
This matches the output of the calculator above.
| Rule | Form | Result | Example |
|---|---|---|---|
| Power rule | a * x^n | n * a * x^(n-1) | d/dx[4x^3] = 12x^2 |
| Constant rule | c | 0 | d/dx[7] = 0 |
| Constant multiple | a * f(x) | a * f'(x) | d/dx[5x^2] = 5 * 2x = 10x |
| Sum rule | f(x) + g(x) | f'(x) + g'(x) | Differentiate each term separately |
The derivative of a function is its instantaneous rate of change. For polynomials, every term is differentiated independently using the power rule, multiplying the coefficient by the exponent and dropping the exponent by one, then the results are combined into a single expression.
For a formal treatment of polynomial derivatives, see Wolfram MathWorld's derivative page and the MIT OpenCourseWare single-variable calculus notes.
This tool covers polynomials. For trig functions (sin, cos), natural log, or exponential functions, the chain rule and product rule also apply. Those topics are covered well at Paul's Online Math Notes (Lamar University).
Differentiation and integration undo each other, and both lean on the same exponent rules.
The derivative of a function measures its instantaneous rate of change at any point. For a polynomial f(x), the derivative f'(x) gives the slope of the tangent line at x. It is the foundation of differential calculus.
The power rule states: d/dx[a * x^n] = n * a * x^(n-1). Multiply the coefficient by the exponent, then reduce the exponent by one. For example, the derivative of 4x^3 is 12x^2.
The derivative of any constant is zero. Constants do not change, so their rate of change is 0. For example, d/dx[7] = 0.
This tool handles polynomial expressions using the power rule. For trig, exponential, or composite functions, see a full symbolic CAS such as Wolfram Alpha or Symbolab.
It shows each term, applies the power rule to that term, and lists the resulting derivative term. The final derivative expression is assembled from all terms.
Marcus Vance spent a decade building data pipelines and testing calculation logic before he started writing about the math behind the tools. On MathCalcTools he checks every formula against the source math, then writes out the steps so a reader can follow the same path a calculator takes.