Numerical Derivative Calculator
Compute the derivative of a polynomial term f(x) = c·xⁿ and evaluate it at a chosen x value using the power rule. Ideal for calculus students needing step-by-step differentiation of single-term expressions.
About this calculator
Differentiation measures how a function's output changes with respect to its input. For a monomial f(x) = c·xⁿ, the power rule gives the derivative: f′(x) = c·n·x^(n−1). The special case where the power is 0 (a constant term) always yields a derivative of 0, since constants have no rate of change. This calculator evaluates: f′(x) = (power === 0) ? 0 : coefficient × power × x_value^(power − 1). The constant term entered separately has no effect on the derivative result. To find the slope of the tangent line at any point, substitute x = x₀ into f′(x). The power rule extends to all real exponents and is the building block for differentiating more complex polynomial and rational functions.
How to use
Find the derivative of f(x) = 5x⁴ + 9 at x = 2. Enter coefficient = 5, power = 4, constant = 9, x = 2. Since power ≠ 0, apply the rule: f′(x) = 5 · 4 · x^(4−1) = 20x³. At x = 2: f′(2) = 20 · (2³) = 20 · 8 = 160. The constant 9 contributes zero. The slope of f(x) at x = 2 is 160.
Frequently asked questions
How does the power rule work for finding derivatives of polynomial terms?
The power rule states that d/dx [c·xⁿ] = c·n·x^(n−1). You multiply the coefficient by the exponent, then reduce the exponent by one. For example, d/dx [7x⁵] = 35x⁴. This rule works for any real number exponent, including fractions and negatives. For a polynomial with multiple terms, apply the rule to each term independently and sum the results.
What happens to the derivative of a constant term in a polynomial?
The derivative of any constant is zero, because a constant does not change as x varies — its slope is always flat. Using the power rule: a constant c = c·x⁰, so d/dx [c·x⁰] = c·0·x^(−1) = 0. This is why the constant field in the calculator does not affect the derivative output. In practical terms, adding or subtracting a constant shifts the graph up or down but does not change its slope at any point.
When should you evaluate a derivative at a specific x value versus leaving it as a general expression?
The general derivative f′(x) is a formula that gives the slope at every point on the curve simultaneously, which is useful for analysis, optimization, and graphing. Evaluating at a specific x = x₀ gives a single number — the exact slope of the tangent line at that point — which is useful when you need the instantaneous rate of change at one moment, such as the speed of an object at a precise time or the marginal cost at a given production level. In optimization, you set f′(x) = 0 and solve for x rather than evaluating at a known point.