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Implicit Differentiation Solver

Compute dy/dx for implicitly defined equations step by step

Select Equation Type

Circle x^2+y^2=r^2 Ellipse x^2/a^2+y^2/b^2=1 Product xy = c

x^2 + y^2 = r^2

x^2 + y^2 =

Implicit Differentiation Method

1. Differentiate both sides with respect to x

2. Apply chain rule: d/dx f(y) = f(y)*dy/dx

3. Collect all dy/dx terms on one side

4. Factor dy/dx and solve for dy/dx

Implicit differentiation allows finding derivatives 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.

⚠ Remember: every time you differentiate a y-term, multiply by dy/dx. For products like xy, use the product rule: d/dx(xy)=y+x*dy/dx.

What Is Implicit Differentiation?

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.

Chain Rule

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.

Product Rule

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).

Solving for dy/dx

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.

Applications

Related rates, tangent lines to implicit curves, inverse function derivatives, optimization with constraints.

Teaching Example: x^2+y^2=25. d/dx: 2x+2y*dy/dx=0. Solve: 2y*dy/dx=-2x -> dy/dx=-x/y. At point (3,4): dy/dx=-3/4.

Applications

Calculus Related Rates Tangent Lines Physics Geometry

Frequently Asked Questions

What is implicit differentiation?▼

Differentiate both sides of an equation, treat y as y(x). Multiply y-terms by dy/dx, then solve for dy/dx.

Why multiply by dy/dx?▼

Chain rule: d/dx f(y(x)) = f(y)*dy/dx. Since y depends on x, differentiation of y-terms requires the chain rule.

Circle dy/dx?▼

x^2+y^2=r^2 -> dy/dx=-x/y. The derivative depends on both x and y since multiple y values correspond to each x.

Product xy derivative?▼

xy=c -> product rule: y+x*dy/dx=0 -> dy/dx=-y/x. The derivative equals -y/x at any point (x,y) on the curve.

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