Textbook Question
If x¹/³ + y¹/³ = 4, find d²y/dx² at the point (8, 8).
4. Applications of Derivatives
Implicit Differentiation
3:29 minutesProblem 3.89
Textbook Question
Find the slope of the curve x³y³ + y² = x + y at the points (1, 1) and (1, -1).
Verified step by step guidance1
To find the slope of the curve at a given point, we need to find the derivative of the curve with respect to x. The curve is given by the equation: x³y³ + y² = x + y.
Use implicit differentiation to differentiate both sides of the equation with respect to x. Remember that y is a function of x, so when differentiating terms involving y, apply the chain rule.
Differentiate the left side: For x³y³, use the product rule: d/dx(x³y³) = x³ * d/dx(y³) + y³ * d/dx(x³). For y², use the chain rule: d/dx(y²) = 2y * dy/dx.
Differentiate the right side: d/dx(x) = 1 and d/dx(y) = dy/dx.
After differentiating, solve the resulting equation for dy/dx to find the expression for the slope of the curve. Substitute the points (1, 1) and (1, -1) into this expression to find the slope at each point.
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Implicit Differentiation
Implicit differentiation is a technique used to find the derivative of a function defined implicitly by an equation involving both x and y. Instead of solving for y explicitly, we differentiate both sides of the equation with respect to x, treating y as a function of x. This allows us to find dy/dx, which represents the slope of the curve at any point.
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Slope of a Curve
The slope of a curve at a given point is defined as the rate of change of the y-coordinate with respect to the x-coordinate at that point. Mathematically, it is represented by the derivative of the function at that point. For curves defined by implicit equations, the slope can be found using the derivative obtained through implicit differentiation.
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Evaluating Derivatives at Specific Points
Once the derivative (dy/dx) is determined, evaluating it at specific points involves substituting the coordinates of those points into the derivative expression. This gives the slope of the curve at those specific locations. In this case, we will substitute the points (1, 1) and (1, -1) into the derived expression to find the respective slopes.
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