Calculate the derivative of a single term polynomial in the form f(x) = axn.
Enter values to see the result
How to Calculate Derivatives: A Comprehensive Guide
In calculus, the derivative represents the rate of change of a function with respect to a variable. Geometrically, it is the slope of the tangent line to the graph of the function at any given point.
The Power Rule
The Power Rule is the most fundamental technique used to calculate derivatives of polynomial functions. The formula is:
d/dx [axn] = n · axn-1
Steps to Calculate a Basic Derivative
Identify the Coefficient (a): This is the number multiplying the variable.
Identify the Exponent (n): This is the power to which the variable is raised.
Multiply: Multiply the exponent by the coefficient (n × a). This becomes your new coefficient.
Subtract: Subtract 1 from the original exponent (n – 1). This becomes your new power.
Example Calculation:
Find the derivative of f(x) = 4x3
1. Multiply exponent (3) by coefficient (4): 3 × 4 = 12.
2. Subtract 1 from the exponent: 3 – 1 = 2.
3. Result: f'(x) = 12x2
Essential Rules of Differentiation
Constant Rule: The derivative of any constant (e.g., 5, 100, π) is always 0.
Sum/Difference Rule: The derivative of a sum is the sum of the derivatives: (f + g)' = f' + g'.
Product Rule: Used when two functions are multiplied: f'g + fg'.
Quotient Rule: Used for fractions: (f'g – fg') / g2.
Chain Rule: Used for composite functions (a function inside another function): dy/dx = dy/du · du/dx.
Common Derivative Table
Function f(x)
Derivative f'(x)
sin(x)
cos(x)
cos(x)
-sin(x)
ex
ex
ln(x)
1/x
function calculateDerivative() {
var a = document.getElementById('coeff').value;
var n = document.getElementById('exponent').value;
var resultDisplay = document.getElementById('calcResult');
if (a === "" || n === "") {
resultDisplay.innerHTML = "Please enter both values.";
resultDisplay.style.color = "#dc3545";
return;
}
var coeff = parseFloat(a);
var exponent = parseFloat(n);
if (isNaN(coeff) || isNaN(exponent)) {
resultDisplay.innerHTML = "Invalid numeric input.";
resultDisplay.style.color = "#dc3545";
return;
}
resultDisplay.style.color = "#495057";
// Logic for Power Rule: d/dx(ax^n) = (a*n)x^(n-1)
var newCoeff = coeff * exponent;
var newExp = exponent – 1;
// Special Case: Exponent is 0 (Derivative of a constant is 0)
if (exponent === 0) {
resultDisplay.innerHTML = "f'(x) = 0";
return;
}
// Special Case: New Exponent is 0 (Result is just the new coefficient)
if (newExp === 0) {
resultDisplay.innerHTML = "f'(x) = " + newCoeff;
return;
}
// Special Case: New Exponent is 1 (Don't show the ^1)
if (newExp === 1) {
resultDisplay.innerHTML = "f'(x) = " + (newCoeff === 1 ? "" : (newCoeff === -1 ? "-" : newCoeff)) + "x";
return;
}
// General Formatting
var coeffStr = newCoeff === 1 ? "" : (newCoeff === -1 ? "-" : newCoeff);
var resultString = "f'(x) = " + coeffStr + "x" + newExp + "";
resultDisplay.innerHTML = resultString;
}