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
The derivative of a function is its instantaneous rate of change. For polynomials, every term is differentiated independently using the power rule, then the results are combined.
| 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 |
This is the default expression in the calculator. Here is each step applied by hand.
This matches the output of the calculator above. For a formal treatment of polynomial derivatives, see Wolfram MathWorld on the derivative 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).
Also on MathCalcTools: the integral calculator (reverse of differentiation) and the exponent calculator.
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.
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