Calc Calculator

Derivative of a Power Function Calculator

function calculateDerivative() { var coefficientInput = document.getElementById("coefficient").value; var exponentInput = document.getElementById("exponent").value; var resultDiv = document.getElementById("derivativeResult"); var a = parseFloat(coefficientInput); var n = parseFloat(exponentInput); if (isNaN(a) || isNaN(n)) { resultDiv.innerHTML = "Please enter valid numbers for both coefficient and exponent."; return; } var newCoefficient = a * n; var newExponent = n – 1; var derivativeString = ""; if (newCoefficient === 0) { derivativeString = "f'(x) = 0"; } else if (newExponent === 0) { derivativeString = "f'(x) = " + newCoefficient; } else if (newExponent === 1) { derivativeString = "f'(x) = " + newCoefficient + "x"; } else { derivativeString = "f'(x) = " + newCoefficient + "x^{" + newExponent + "}"; } resultDiv.innerHTML = "

Result:

If f(x) = " + a + "x^{" + n + "}, then its derivative is:" + derivativeString + ""; } .calc-container { background-color: #f9f9f9; border: 1px solid #ddd; padding: 20px; border-radius: 8px; max-width: 600px; margin: 20px auto; font-family: 'Segoe UI', Tahoma, Geneva, Verdana, sans-serif; } .calc-container h2 { color: #333; text-align: center; margin-bottom: 20px; } .calc-input-group { margin-bottom: 15px; } .calc-input-group label { display: block; margin-bottom: 5px; color: #555; font-weight: bold; } .calc-input-group input[type="number"] { width: calc(100% – 22px); padding: 10px; border: 1px solid #ccc; border-radius: 4px; box-sizing: border-box; } .calc-button { background-color: #007bff; color: white; padding: 12px 20px; border: none; border-radius: 4px; cursor: pointer; font-size: 16px; width: 100%; display: block; margin-top: 20px; } .calc-button:hover { background-color: #0056b3; } .calc-result { margin-top: 25px; padding: 15px; background-color: #e9ecef; border: 1px solid #dee2e6; border-radius: 4px; text-align: center; } .calc-result h3 { color: #333; margin-top: 0; margin-bottom: 10px; } .calc-result p { margin: 5px 0; color: #333; font-size: 1.1em; } .calc-result .highlight { font-size: 1.3em; font-weight: bold; color: #28a745; } .calc-result .error { color: #dc3545; font-weight: bold; }

Understanding the Derivative of a Power Function

Calculus is a branch of mathematics focused on rates of change and accumulation. One of its fundamental concepts is the derivative, which measures how a function changes as its input changes. In simpler terms, the derivative of a function at a certain point gives you the slope of the tangent line to the function's graph at that point.

What is a Power Function?

A power function is a type of polynomial function that can be expressed in the form f(x) = axn, where:

• a is the coefficient, a constant real number.
• x is the variable.
• n is the exponent, a constant real number.

Examples of power functions include 3x2, 5x-1 (which is 5/x), or even 2x (where n=1) and 7 (where n=0, as 7x0 = 7*1 = 7).

The Power Rule for Differentiation

To find the derivative of a power function, we use a straightforward rule known as the Power Rule. If you have a function f(x) = axn, its derivative, denoted as f'(x) (read as "f prime of x"), is given by the formula:

f'(x) = (a * n)x(n - 1)

Let's break down what this means:

1. You multiply the original coefficient (a) by the original exponent (n). This gives you the new coefficient.
2. You subtract 1 from the original exponent (n). This gives you the new exponent.

Examples:

Let's apply the Power Rule to a few examples:

1. Function: f(x) = 3x2
   • Here, a = 3 and n = 2.
   • New coefficient: 3 * 2 = 6
   • New exponent: 2 - 1 = 1
   • Derivative: f'(x) = 6x1 = 6x
2. Function: f(x) = 5x4
   • Here, a = 5 and n = 4.
   • New coefficient: 5 * 4 = 20
   • New exponent: 4 - 1 = 3
   • Derivative: f'(x) = 20x3
3. Function: f(x) = 7x (which is 7x1)
   • Here, a = 7 and n = 1.
   • New coefficient: 7 * 1 = 7
   • New exponent: 1 - 1 = 0
   • Derivative: f'(x) = 7x0 = 7 * 1 = 7

   This shows that the derivative of a linear function (ax) is simply its coefficient (a).

4. Function: f(x) = 10 (which is 10x0)
   • Here, a = 10 and n = 0.
   • New coefficient: 10 * 0 = 0
   • New exponent: 0 - 1 = -1
   • Derivative: f'(x) = 0x-1 = 0

   This demonstrates that the derivative of any constant function is always zero, as a constant value does not change.

Using the Calculator

Our Derivative of a Power Function Calculator simplifies this process. Simply input the coefficient (a) and the exponent (n) of your power function axn, and the calculator will instantly provide its derivative f'(x) using the Power Rule.