Math Problem Statement
Solution
We are tasked with finding the equation of the tangent line to the ellipse defined by the equation:
at the point using implicit differentiation.
Step 1: Differentiate implicitly with respect to .
We start by differentiating the given equation implicitly. For each term:
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:
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: Here, we use the product rule for differentiation: Simplifying:
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: Again, using the chain rule:
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The constant 52: Since the derivative of a constant is zero:
Thus, the derivative of the entire equation becomes:
Step 2: Substitute the given point .
Substitute and into the equation to simplify:
Simplifying each term: Distribute the : Combine like terms: Solving for :
Thus, the slope of the tangent line is .
Step 3: Use point-slope form to write the equation of the tangent line.
The point-slope form of a line is given by: Substitute and : Multiply through by 7 to eliminate the fraction: Finally, rearrange to get the equation of the tangent line:
Thus, the equation of the tangent line is:
Would you like further details on any part of the solution or have any questions?
Related Questions:
- How does implicit differentiation differ from explicit differentiation?
- Why do we use the product rule for the term ?
- What is the significance of the slope in the context of the tangent line?
- Can we apply this method to any implicit equation to find tangent lines?
- What would the equation of the normal line be at the same point?
Tip:
Always check that the point given lies on the curve by substituting it into the original equation before proceeding with implicit differentiation.
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Math Problem Analysis
Mathematical Concepts
Implicit Differentiation
Ellipses
Tangent Line
Product Rule
Chain Rule
Formulas
Product rule for derivatives: d(uv)/dx = u'v + uv'
Chain rule: d/dx[f(g(x))] = f'(g(x)) * g'(x)
Point-slope form of a line: y - y1 = m(x - x1)
Theorems
Implicit Differentiation Theorem
Equation of a Tangent Line
Suitable Grade Level
Undergraduate Level (Calculus)
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