The slope of a tangent line to a curve at a specific point is one of the fundamental concepts of calculus. It represents the instantaneous rate of change of the function at that exact point. To find this slope, you must use the derivative of the function.
The Step-by-Step Formula
Identify the Function: Let the function be f(x).
Find the Derivative: Calculate f'(x). For a basic power function like f(x) = axⁿ, the derivative is f'(x) = anxⁿ⁻¹.
Plug in the x-value: Substitute the specific x-coordinate (let's call it a) into the derivative to find the slope (m). So, m = f'(a).
Find the y-coordinate: Plug the x-value back into the original function to get y₁ = f(a).
Form the Equation: Use the point-slope form: y – y₁ = m(x – a).
Practical Example
Suppose you have the function f(x) = 2x² + 3x and you want to find the slope of the tangent line at x = 2.
Step 1: Derivative f'(x) = 4x + 3.
Step 2: Substitute x = 2: f'(2) = 4(2) + 3 = 11.
Result: The slope of the tangent line at x = 2 is 11.
Equation: Since f(2) = 2(2)² + 3(2) = 14, the point is (2, 14). The equation is y – 14 = 11(x – 2) or y = 11x – 8.
function calculateTangentLine() {
var a = parseFloat(document.getElementById('coeffA').value);
var n = parseFloat(document.getElementById('powerN').value);
var b = parseFloat(document.getElementById('coeffB').value);
var c = parseFloat(document.getElementById('constC').value);
var xVal = parseFloat(document.getElementById('pointX').value);
if (isNaN(a) || isNaN(n) || isNaN(b) || isNaN(c) || isNaN(xVal)) {
alert("Please enter valid numbers in all fields.");
return;
}
// f(x) = ax^n + bx + c
var yVal = (a * Math.pow(xVal, n)) + (b * xVal) + c;
// f'(x) = (a*n)x^(n-1) + b
var slope = (a * n * Math.pow(xVal, n – 1)) + b;
// Point-slope: y – yVal = slope(x – xVal)
// y = slope*x – slope*xVal + yVal
var intercept = (-slope * xVal) + yVal;
var interceptSign = intercept >= 0 ? "+ " + intercept.toFixed(2) : "- " + Math.abs(intercept).toFixed(2);
document.getElementById('tangentResult').style.display = 'block';
document.getElementById('calcSlope').innerHTML = "Slope (m): " + slope.toFixed(2);
document.getElementById('calcPoint').innerHTML = "Point of Tangency: (" + xVal + ", " + yVal.toFixed(2) + ")";
document.getElementById('calcEquation').innerHTML = "Equation: y = " + slope.toFixed(2) + "x " + interceptSign;
}