Enter the coefficients and exponents for your polynomial expression. This calculator applies the Power Rule: d/dx(axn) = anxn-1.
x
x
x0
Resulting Derivative f'(x):
Understanding Derivatives in Calculus
A derivative measures the sensitivity to change of a function value with respect to a change in its argument. In simpler terms, it represents the instantaneous rate of change or the slope of the tangent line at any given point on a curve.
The Power Rule Explained
The Power Rule is one of the most fundamental tools in differentiation. It states that for any term in the form f(x) = axn, its derivative is:
f'(x) = n · axn-1
Common Rules for Differentiation
Constant Rule: The derivative of a constant (like 7 or 100) is always 0.
Linear Rule: The derivative of ax is simply a.
Sum Rule: The derivative of a sum of functions is the sum of their derivatives.
Practical Example
Consider the function: f(x) = 4x3 + 2x2 – 5
Apply power rule to 4x3: (3 × 4)x3-1 = 12x2
Apply power rule to 2x2: (2 × 2)x2-1 = 4x
Derivative of constant -5: 0
Final Derivative: f'(x) = 12x2 + 4x
Why Use This Calculator?
Our tool helps students, engineers, and physicists quickly find derivatives for complex polynomials without manual calculation errors. It provides the final expression instantly, allowing for faster verification of homework and research data.
function calculateDerivative() {
var resultArea = document.getElementById("derivative-result-area");
var resultDisplay = document.getElementById("result-display");
var explanationDisplay = document.getElementById("explanation-display");
var terms = [];
var steps = [];
// Process Term 1
var c1 = parseFloat(document.getElementById("coeff1").value) || 0;
var e1 = parseFloat(document.getElementById("exp1").value) || 0;
// Process Term 2
var c2 = parseFloat(document.getElementById("coeff2").value) || 0;
var e2 = parseFloat(document.getElementById("exp2").value) || 0;
// Process Term 3 (Constant)
var c3 = parseFloat(document.getElementById("coeff3").value) || 0;
var e3 = 0;
function differentiate(c, e) {
if (e === 0) return { coeff: 0, exp: 0, str: "" };
var newCoeff = c * e;
var newExp = e – 1;
var termStr = "";
if (newCoeff === 0) return { coeff: 0, exp: 0, str: "" };
// Handle coefficient string
var sign = newCoeff > 0 ? " + " : " – ";
var absCoeff = Math.abs(newCoeff);
if (newExp === 0) {
termStr = absCoeff.toString();
} else if (newExp === 1) {
termStr = (absCoeff === 1 ? "" : absCoeff) + "x";
} else {
termStr = (absCoeff === 1 ? "" : absCoeff) + "x" + newExp + "";
}
return { coeff: newCoeff, exp: newExp, str: termStr, sign: sign };
}
var d1 = differentiate(c1, e1);
var d2 = differentiate(c2, e2);
var d3 = differentiate(c3, e3);
var finalString = "";
var hasTerms = false;
// Build the string properly
var derivativeArray = [d1, d2, d3];
for (var i = 0; i < derivativeArray.length; i++) {
var d = derivativeArray[i];
if (d.str !== "") {
if (!hasTerms) {
// First term – handle negative sign but skip leading plus
var prefix = d.coeff < 0 ? "-" : "";
finalString += prefix + d.str;
} else {
finalString += d.sign + d.str;
}
hasTerms = true;
}
}
if (!hasTerms) {
finalString = "0";
}
resultDisplay.innerHTML = "f'(x) = " + finalString;
explanationDisplay.innerHTML = "Applied d/dx to individual terms: " +
(c1 !== 0 ? "(" + c1 + "·" + e1 + ")x" + (e1-1) + "" : "0") + " + " +
(c2 !== 0 ? "(" + c2 + "·" + e2 + ")x" + (e2-1) + "" : "0") + " + 0″;
resultArea.style.display = "block";
}