Compute Derivatives and Integrals for functions of the form f(x) = axⁿ
Original Function:
The Derivative (f'(x)):
Instantaneous Rate of Change at x:
The Indefinite Integral (∫f(x)dx):
Understanding Calculus Basics
Calculus is the mathematical study of continuous change. It is divided into two main branches: Differential Calculus (concerning rates of change and slopes of curves) and Integral Calculus (concerning accumulation of quantities and areas under curves).
The Power Rule for Derivatives
One of the most fundamental rules in calculus is the Power Rule. It states that if you have a function f(x) = axⁿ, the derivative is calculated by multiplying the coefficient by the exponent and subtracting one from the exponent:
f'(x) = (a * n)xⁿ⁻¹
For example, if you have f(x) = 3x², the derivative is f'(x) = 6x¹ or simply 6x. This represents the slope of the tangent line at any given point x.
The Power Rule for Integrals
Integration is essentially the reverse process of differentiation. To find the indefinite integral of a power function, you add one to the exponent and divide the coefficient by the new exponent:
∫axⁿ dx = [a / (n + 1)]xⁿ⁺¹ + C
Note: This rule applies for all values of n except n = -1 (which results in a natural logarithm).
Practical Example
Imagine an object moving where its position is defined by the function p(t) = 4t³. To find its velocity at 2 seconds:
Integral (Accumulation): ∫4t³ dt = (4/4)t⁴ = t⁴ + C
function calculateCalculus() {
var a = parseFloat(document.getElementById("coefficientA").value);
var n = parseFloat(document.getElementById("exponentN").value);
var x = parseFloat(document.getElementById("evalX").value);
var resultsDiv = document.getElementById("results");
if (isNaN(a) || isNaN(n)) {
alert("Please enter valid numbers for the coefficient and exponent.");
return;
}
// 1. Original Function Display
document.getElementById("originalFunc").innerHTML = a + "x" + n + "";
// 2. Derivative Calculation f'(x) = (a*n)x^(n-1)
var derCoeff = a * n;
var derExp = n – 1;
var derStr = derCoeff.toFixed(2) + "x" + derExp.toFixed(2) + "";
document.getElementById("derivativeFunc").innerHTML = derStr;
// 3. Evaluation of Derivative
if (!isNaN(x)) {
var val = derCoeff * Math.pow(x, derExp);
document.getElementById("derivativeEval").innerText = "At x = " + x + ", f'(x) = " + val.toLocaleString(undefined, {maximumFractionDigits: 4});
} else {
document.getElementById("derivativeEval").innerText = "Enter x value to see evaluation.";
}
// 4. Integral Calculation integral = (a/(n+1))x^(n+1)
if (n === -1) {
document.getElementById("integralFunc").innerHTML = a + " ln|x| + C";
} else {
var intCoeff = a / (n + 1);
var intExp = n + 1;
var intStr = intCoeff.toFixed(2) + "x" + intExp.toFixed(2) + " + C";
document.getElementById("integralFunc").innerHTML = intStr;
}
resultsDiv.style.display = "block";
}