Find dy/dx by implicit differentiation with step-by-step chain rule work
Implicit differentiation allows finding dy/dx when y is not expressed explicitly in terms of x. The key is treating y as a function of x and applying the chain rule to every term containing y. This is the method used when you need to find dy/dx by implicit differentiation.
Differentiate both sides with respect to x, attach dy/dx to every derivative involving y, move the dy/dx terms to one side, factor dy/dx, and divide by its coefficient. For x^2 + y^2 = r^2, this gives 2x + 2y dy/dx = 0, so dy/dx = -x/y.
Implicit differentiation finds the derivative of functions defined implicitly by equations. Instead of solving for y first, differentiate term by term. Every y-term gets a dy/dx factor from the chain rule. Finally, solve for dy/dx algebraically.
d/dx f(y) = f(y)*dy/dx. This is the key step. Examples: d/dx(y^2)=2y*y, d/dx(sin(y))=cos(y)*y.
For xy: d/dx(xy)=1*y+x*1*dy/dx=y+x*dy/dx. For x^2y: use product rule (u=x^2, v=y).
After differentiation, collect all dy/dx terms. Factor out dy/dx and divide by the coefficient. Result is dy/dx in terms of x and y.
Related rates, tangent lines to implicit curves, inverse function derivatives, optimization with constraints.
| Expression | Derivative Rule | Why |
|---|---|---|
| y^2 | 2y dy/dx | Chain rule because y depends on x. |
| xy | y + x dy/dx | Product rule plus chain rule. |
| sin(y) | cos(y) dy/dx | Outer derivative times derivative of y. |
| x^2 + y^2 = r^2 | dy/dx = -x/y | Standard circle derivative. |
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