Implicit differemation 3) \( \frac{x^{2}}{y}=\cos \left(x^{2}\right) \)

To tackle the equation \( \frac{x^{2}}{y}=\cos \left(x^{2}\right) \) using implicit differentiation, we first rewrite it in a more manageable form. Multiply both sides by \( y \) to get \( x^2 = y \cos(x^2) \). Now, differentiate both sides with respect to \( x \). On the left side, you get \( 2x \), and on the right side, you’ll have to use the product rule, leading to \( \cos(x^2) \frac{dy}{dx} - 2x y \sin(x^2) \). Setting both sides equal will provide a good path to find \( \frac{dy}{dx} \). After finding \( \frac{dy}{dx} \), don't forget to isolate it to clearly express it in terms of \( x \) and \( y \). This implicit differentiation technique is especially handy when dealing with curves defined by equations not easily solvable for \( y \). So, keep that in mind next time you encounter a complicated equation! And remember, one common mistake is forgetting to apply the chain rule while differentiating terms involving \( y \). That’s a surefire way to trip yourself up, so double-check for any sneaky \( \frac{dy}{dx} \) terms hiding out!