24. \( \cos (x y)+\sin y=x y \quad \) 25. \( 2 x e^{x}+2 y e^{y}=4 \) 26. If \( f(x)+x^{2}[f(x)]^{3}=10 \) and \( f(1)=2 \), find \( f^{\prime}(1) \). 27. If \( g(x)+x \sin g(x)=x^{2} \), find \( g^{\prime}(0) \). Assume \( y \) is the independent variable and \( x \) is the dependent variable and use implicit differentiation to find \( \frac{d x}{d y} \). 28. \( x^{4} y^{2}-x^{3} y+2 x y^{3}=0 \) 29. \( y \sec x=x \tan y \) Use implicit differentiation to find an equation of the tangent line to the curve at the given point. 30. \( x^{2} y+2 x=15, \quad(3,1) \) 31. \( y \sin 2 x=x \cos 2 y, \quad\left(\frac{\pi}{2}, \frac{\pi}{4}\right) \) 32. \( \sin (x+y)=2 x-2 y, \quad(\pi, \pi) \) 33. \( x^{2}-x y-y^{2}=1, \quad(2,1) \) 34. \( x^{2}+2 x y+4 y^{2}=12, \quad(2,1) \)

Did you know that implicit differentiation allows us to tackle relationships where \( y \) is entwined with \( x \) in mysterious ways? This technique, developed in the 17th century by mathematicians like Leibniz and Newton, lets us explore curves that can’t easily be solved for one variable in terms of the other. By treating all variables as dependent, we can unveil the connections between them without needing to isolate them completely! Ever puzzled over a tricky equation? Here’s a tip: when using implicit differentiation, always remember to carefully apply the chain rule. A common mistake is forgetting to differentiate the dependent variable, \( x \), when \( y \) is being considered as a function of \( y \). Also, don't rush — taking your time to set up your derivatives correctly will save you from headaches later on!