Textbook Question
60–62. {Use of Tech} Multiple tangent lines Complete the following steps. <IMAGE>
a. Find equations of all lines tangent to the curve at the given value of x.
x+y³−y=1; x=1
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4. Applications of Derivatives
Implicit Differentiation
2:55 minutesProblem 3.8.63a
Textbook Question
Witch of Agnesi Let y(x²+4)=8 (see figure). <IMAGE>
a. Use implicit differentiation to find dy/dx.
Verified step by step guidance1
Start by differentiating both sides of the equation y(x²+4)=8 with respect to x. This involves using implicit differentiation, where you treat y as a function of x.
Apply the product rule to differentiate the left side of the equation. The product rule states that d(uv)/dx = u(dv/dx) + v(du/dx), where u and v are functions of x. Here, u = y and v = (x²+4).
Differentiate u = y with respect to x, which gives dy/dx. Differentiate v = (x²+4) with respect to x, which gives 2x.
Substitute the derivatives back into the product rule: y(2x) + (x²+4)(dy/dx) = 0. This equation comes from differentiating the left side of the original equation.
Solve for dy/dx by isolating it on one side of the equation. Rearrange the terms to get (x²+4)(dy/dx) = -y(2x), and then divide both sides by (x²+4) to find dy/dx = -y(2x)/(x²+4).
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Video duration:
2mKey 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 differentiate equations where the dependent and independent variables are not isolated on one side. Instead of solving for y explicitly, we differentiate both sides of the equation with respect to x, applying the chain rule when necessary. This method is particularly useful for curves defined by equations that cannot be easily rearranged.
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Finding The Implicit Derivative
Chain Rule
The chain rule is a fundamental principle in calculus that allows us to differentiate composite functions. It states that if a function y is composed of another function u, then the derivative of y with respect to x can be found by multiplying the derivative of y with respect to u by the derivative of u with respect to x. This is essential in implicit differentiation, where y is often a function of x indirectly.
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Witch of Agnesi
The Witch of Agnesi is a specific type of curve defined by the equation y(x² + 4) = 8, which can be rearranged to express y in terms of x. This curve is notable in mathematics for its unique shape and properties, resembling a bell. Understanding its equation and characteristics is crucial for applying differentiation techniques effectively in the context of the problem.
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