f'(x): The derivative of your function — a new expression that gives the slope (rate of change) of f(x) at any point x. Plug in a specific x value to get the slope of the tangent line there.
Steps: Each line shows one term being differentiated. For a·xn, the power rule produces (a·n)·xn−1. Constants become zero because they have no rate of change.
Zero derivative: If f'(x) = 0, the original function is a constant — its graph is a flat horizontal line with no slope anywhere.
Practical use: Setting f'(x) = 0 and solving finds critical points — where the function has local maxima, minima, or inflection points.
How This Calculator Works
You type a polynomial in x (for example, 3x^2 + 2x + 1). The tool parses it into individual terms and applies the power rule — multiply the coefficient by the exponent, then reduce the exponent by one. It handles integer and decimal coefficients but does not support trigonometric, exponential, or logarithmic functions.
Quick Questions
What is the power rule?
The power rule states that the derivative of a·xn is (a·n)·xn−1. It is the most fundamental differentiation rule for polynomials and works for any real exponent.
Can this handle sin(x), e^x, or ln(x)?
No. This calculator only supports polynomial terms (powers of x with numeric coefficients). For transcendental functions you would need the chain rule, product rule, or quotient rule — which are beyond this tool's scope.
What does a negative derivative mean?
A negative value of f'(x) at a given x means the original function is decreasing at that point — its graph slopes downward from left to right. The more negative the value, the steeper the decline.
How do I find the second derivative?
Take the result f'(x) and enter it back into the calculator as a new function. The output will be f''(x), which tells you about the concavity — whether the curve bends upward or downward.