Implicit Differentiation

Ever tried to solve for y in a complex equation like x²+y²=1? Sometimes it's impossible or too complex. That's when implicit differentiation comes to the rescue! This technique lets you find the derivative without isolating y.

The key is to remember that whenever you differentiate y with respect to x, it becomes dy/dx. When differentiating terms with y variables, use the chain rule by multiplying by dy/dx. After differentiating both sides of the equation, gather all the dy/dx terms to one side and solve.

Let's see an example: For xy + y = 8x, we differentiate both sides:

• Left side: y + x$dy/dx$ + $dy/dx$ = y + $x+1$$dy/dx$
• Right side: 8
• Solving for dy/dx: $x+1$$dy/dx$ = 8-y, so dy/dx = $8-y$/$x+1$

Pro Tip: When finding the second derivative $d²y/dx²$, first find dy/dx, then differentiate again using the quotient rule. It gets complex, but breaking it down step-by-step makes it manageable!

For second derivatives, like in x² + xy² + y³ = 27, you'll first find dy/dx, then differentiate again. Remember to substitute your expression for dy/dx into the final equation for d²y/dx².