Implicit Derivative Calculator

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Implicit Derivative Evaluator for x² + y² = R²

Enter the coordinates (x, y) of a point on the circle to find the slope of the tangent line (dy/dx) at that point.

Result:

function calculateImplicitDerivative() { var x_coordinate = parseFloat(document.getElementById("x_coordinate").value); var y_coordinate = parseFloat(document.getElementById("y_coordinate").value); var resultDiv = document.getElementById("result"); if (isNaN(x_coordinate) || isNaN(y_coordinate)) { resultDiv.innerHTML = "Please enter valid numbers for both X and Y coordinates."; return; } if (y_coordinate === 0) { resultDiv.innerHTML = "At y = 0, the tangent line is vertical, and dy/dx is undefined."; } else { var dy_dx = -x_coordinate / y_coordinate; resultDiv.innerHTML = "For the equation x² + y² = R², at the point (" + x_coordinate + ", " + y_coordinate + "), dy/dx = " + dy_dx.toFixed(4) + "."; } }

Understanding Implicit Differentiation

Implicit differentiation is a powerful technique in calculus used to find the derivative of a function when it's not explicitly defined as y = f(x). Instead, y is defined implicitly by an equation relating x and y.

When to Use Implicit Differentiation?

You typically use implicit differentiation when it's difficult or impossible to solve an equation for y in terms of x. Common examples include equations of circles, ellipses, or more complex curves where y is intertwined with x.

The Process

  1. Differentiate Both Sides: Take the derivative of both sides of the equation with respect to x.
  2. Apply Chain Rule: When differentiating terms involving y, remember to apply the chain rule. For example, the derivative of y² with respect to x is 2y * (dy/dx).
  3. Isolate dy/dx: After differentiating, rearrange the equation to solve for dy/dx.

Example: The Circle x² + y² = R²

Let's consider the equation of a circle centered at the origin: x² + y² = R², where R is the radius. We want to find dy/dx.

  1. Differentiate both sides with respect to x: d/dx(x²) + d/dx(y²) = d/dx(R²)
  2. Apply the power rule and chain rule: 2x + 2y * (dy/dx) = 0 (Since R is a constant, R² is also a constant, and its derivative is 0)
  3. Isolate dy/dx: 2y * (dy/dx) = -2x dy/dx = -2x / (2y) dy/dx = -x / y

This formula tells us the slope of the tangent line at any point (x, y) on the circle (as long as y ≠ 0).

How This Calculator Works

This calculator uses the derived formula dy/dx = -x / y for the circle x² + y² = R². You input the x and y coordinates of a point on the circle, and it will compute the value of the derivative at that specific point. Note that if y = 0, the tangent line is vertical, and the derivative is undefined.

Realistic Example:

Consider a circle with equation x² + y² = 25 (so R=5). Let's find the slope of the tangent at the point (3, 4).

  • x = 3
  • y = 4
  • Using the formula dy/dx = -x / y:
  • dy/dx = -3 / 4 = -0.75

This means at the point (3, 4) on the circle, the tangent line has a slope of -0.75.

Now, try it yourself with the calculator above!

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