A few implicit differentiation problems

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SUMMARY

This discussion focuses on solving implicit differentiation problems, specifically finding the equations of tangent lines for various curves. Key examples include determining the tangent line for the curve defined by the equation xy3 + xy = 16 at the point (8,1) and evaluating the derivative for the equation (6x - y)4 + 2y3 = 2399 at (1,-1). The discussion also covers finding slopes for a lemniscate curve and other implicit functions, emphasizing the necessity of showing work to receive assistance.

PREREQUISITES
  • Understanding of implicit differentiation techniques
  • Familiarity with tangent line equations in the form y = mx + b
  • Knowledge of evaluating derivatives at specific points
  • Basic algebraic manipulation skills
NEXT STEPS
  • Practice implicit differentiation with various polynomial and transcendental functions
  • Learn how to derive tangent line equations from implicit functions
  • Explore advanced topics in calculus, such as higher-order derivatives
  • Study the properties of lemniscates and their applications in calculus
USEFUL FOR

Students and educators in calculus, particularly those focusing on implicit differentiation and tangent line analysis, as well as anyone seeking to improve their problem-solving skills in advanced mathematics.

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1.Use implicit differentiation to find the equation of the tangent line to the curve
xy^3+xy=16 at the point (8,1) . The equation of this tangent line can be written in the form y=mx+b

2.For the equation given below, evaluate y^1 at the point(1,-1) .
(6x-y)^4+2y^3=2399.

3.Find the equation of the tangent line to the curve (a lemniscate)
2(x^2+y^2)^2=25(x^2-y^2) at the point (3,-1). The equation of this tangent line can be written in the form y=mx+b

4.Find the slope of the tangent line to the curve sqrt(3x+3y)+sqrt(xy)=6.48 at the point
(1,5).

5.Use implicit differentiation to find the slope of the tangent line to the curve
y/(x+4y)=x^7+2 at the point (1,-3/11).

Anyone who can help... it would be greatly appreciated! :confused:
 
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