Textbook Question
In Exercises 53 and 54, find both dy/dx (treating y as a differentiable function of x) and dx/dy (treating x as a differentiable function of y). How do dy/dx and dx/dy seem to be related?
53. xy³ + x²y = 6
4. Applications of Derivatives
Implicit Differentiation
7:55 minutesProblem 3.7.47
Textbook Question
Theory and Examples
Intersecting normal line The line that is normal to the curve x² + 2xy – 3y² = 0 at (1,1) intersects the curve at what other point?
Verified step by step guidance1
First, find the derivative of the given implicit function \(x^2 + 2xy - 3y^2 = 0\) with respect to \(x\) using implicit differentiation. This will help us find the slope of the tangent line at any point \((x, y)\) on the curve.
Differentiate each term: \(\frac{d}{dx}(x^2) = 2x\), \(\frac{d}{dx}(2xy) = 2y + 2x\frac{dy}{dx}\), and \(\frac{d}{dx}(-3y^2) = -6y\frac{dy}{dx}\). Set the derivative equal to zero: \(2x + 2y + 2x\frac{dy}{dx} - 6y\frac{dy}{dx} = 0\).
Solve for \(\frac{dy}{dx}\) to find the slope of the tangent line: \(\frac{dy}{dx} = \frac{2x + 2y}{6y - 2x}\). Evaluate this derivative at the point \((1, 1)\) to find the slope of the tangent line at that point.
The slope of the normal line is the negative reciprocal of the tangent slope. Calculate this slope and use the point-slope form of a line equation \(y - y_1 = m(x - x_1)\) to find the equation of the normal line at \((1, 1)\).
Substitute the equation of the normal line into the original curve equation \(x^2 + 2xy - 3y^2 = 0\) to find the other point of intersection. Solve the resulting system of equations to find the coordinates of the other intersection point.
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Normal Line to a Curve
A normal line to a curve at a given point is perpendicular to the tangent line at that point. To find the equation of the normal line, first determine the slope of the tangent line using derivatives, then use the negative reciprocal of this slope for the normal line. This concept is crucial for understanding how the normal line interacts with the curve.
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Implicit Differentiation
Implicit differentiation is used when dealing with equations where y is not isolated. It involves differentiating both sides of the equation with respect to x, treating y as a function of x. This technique is essential for finding the derivative of the given curve x² + 2xy – 3y² = 0, which helps in determining the slope of the tangent line at a specific point.
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Intersection of Curves
Finding the intersection of curves involves solving the equations of the curves simultaneously. After determining the equation of the normal line, substitute it back into the original curve equation to find other points of intersection. This concept is key to identifying where the normal line intersects the curve again, beyond the initial point of tangency.
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