Calculate the derivative dy/dx for equations in the form: axn + bym + cxy + dx + ey + k = 0
Evaluate at Point (Optional)
Result:
Solution Steps:
Differentiate each term with respect to x.
Apply the Chain Rule to terms containing y: d/dx[f(y)] = f'(y) * dy/dx.
Group all terms containing dy/dx on one side.
Factor out dy/dx and solve for it.
Understanding Implicit Differentiation
In standard calculus, we usually deal with explicit functions like y = f(x). However, many mathematical relationships are defined implicitly, such as the equation of a circle x² + y² = 25. In these cases, y cannot be easily isolated on one side of the equation.
The Methodology
Implicit differentiation allows us to find the derivative dy/dx without solving for y first. The core principle is treating y as a function of x, y(x), and applying the Chain Rule whenever we differentiate a term containing y.
The general formula for the derivative of an implicit function F(x, y) = 0 is:
dy/dx = – ( ∂F/∂x ) / ( ∂F/∂y )
Where ∂F/∂x is the partial derivative of the function with respect to x, and ∂F/∂y is the partial derivative with respect to y.
Example Walkthrough
Consider the equation: x² + y² = 25
Differentiate with respect to x: d/dx(x²) + d/dx(y²) = d/dx(25)
Apply power rule and chain rule: 2x + 2y(dy/dx) = 0
Isolate the derivative term: 2y(dy/dx) = -2x
Solve for dy/dx: dy/dx = -2x / 2y = -x/y
Why Use This Calculator?
This tool is essential for students in Calculus I or Calculus II who are dealing with complex implicit equations. It handles powers, products of x and y, and constant terms, providing the algebraic derivative and the numerical slope at any given point (x, y) on the curve.
function calculateImplicitDerivative() {
// Get values from inputs
var a = parseFloat(document.getElementById('coeff_a').value) || 0;
var n = parseFloat(document.getElementById('pow_n').value) || 0;
var b = parseFloat(document.getElementById('coeff_b').value) || 0;
var m = parseFloat(document.getElementById('pow_m').value) || 0;
var c = parseFloat(document.getElementById('coeff_c').value) || 0;
var d = parseFloat(document.getElementById('coeff_d').value) || 0;
var e = parseFloat(document.getElementById('coeff_e').value) || 0;
var valX = document.getElementById('eval_x').value;
var valY = document.getElementById('eval_y').value;
// Partial derivative wrt x (Fx)
// d/dx [ax^n + by^m + cxy + dx + ey + k]
// = (a*n)x^(n-1) + cy + d
// Construct Fx string components
var fx_part1 = (a * n) !== 0 ? (a * n) + "x" + (n – 1 !== 1 ? "" + (n – 1) + "" : "") : "";
if (n-1 === 0 && (a*n) !== 0) fx_part1 = (a*n);
var fx_part2 = c !== 0 ? (c > 0 ? " + " : " – ") + Math.abs(c) + "y" : "";
var fx_part3 = d !== 0 ? (d > 0 ? " + " : " – ") + Math.abs(d) : "";
var fx_str = fx_part1 + fx_part2 + fx_part3;
if (fx_str === "") fx_str = "0";
// Partial derivative wrt y (Fy)
// d/dy [ax^n + by^m + cxy + dx + ey + k]
// = (b*m)y^(m-1) + cx + e
var fy_part1 = (b * m) !== 0 ? (b * m) + "y" + (m – 1 !== 1 ? "" + (m – 1) + "" : "") : "";
if (m-1 === 0 && (b*m) !== 0) fy_part1 = (b*m);
var fy_part2 = c !== 0 ? (c > 0 ? " + " : " – ") + Math.abs(c) + "x" : "";
var fy_part3 = e !== 0 ? (e > 0 ? " + " : " – ") + Math.abs(e) : "";
var fy_str = fy_part1 + fy_part2 + fy_part3;
if (fy_str === "") fy_str = "0";
// Display formula dy/dx = -Fx / Fy
var resultDiv = document.getElementById('result-area');
var formulaDisplay = document.getElementById('formula-display');
var slopeDisplay = document.getElementById('slope-display');
resultDiv.style.display = "block";
formulaDisplay.innerHTML = "dy/dx = – (" + fx_str + ") / (" + fy_str + ")";
// Numerical Evaluation
if (valX !== "" && valY !== "") {
var x = parseFloat(valX);
var y = parseFloat(valY);
var numFx = (a * n * Math.pow(x, n – 1)) + (c * y) + d;
var numFy = (b * m * Math.pow(y, m – 1)) + (c * x) + e;
if (numFy === 0) {
slopeDisplay.innerHTML = "Slope at (" + x + ", " + y + "): Undefined (Vertical Tangent)";
slopeDisplay.style.color = "#c0392b";
} else {
var slope = – (numFx / numFy);
slopeDisplay.innerHTML = "Slope at (" + x + ", " + y + "): " + slope.toFixed(4);
slopeDisplay.style.color = "#27ae60";
}
} else {
slopeDisplay.innerHTML = "";
}
// Smooth scroll to result
resultDiv.scrollIntoView({ behavior: 'smooth', block: 'nearest' });
}