Implicit Differentiation Formula

Implicit differentiation is finding dy/dx when y is defined implicitly by an equation like F(x, y) = 0, by differentiating both sides and solving for dy/dx.

The Formula

For F(x,y)=0: differentiate both sides with respect to x, apply chain rule to y-terms, solve for dydx.

When to use: Sometimes you can't (or don't want to) solve for y explicitly. Instead, differentiate the whole equation as-is. Every time you differentiate a y-term, attach dydx by the chain rule (since y secretly depends on x), then solve for dydx.

Quick Example

Find dydx for the circle x2+y2=25.
Differentiate: 2x+2ydydx=0.
Solve: dydx=−xy
At (3,4): slope =−34.

Notation

dydx found implicitly. Alternatively, dydx=−FxFy where Fx and Fy are partial derivatives of F(x,y).

What This Formula Means

Finding dydx when y is defined implicitly by an equation like F(x,y)=0, by differentiating both sides and solving for dydx.

Sometimes you can't (or don't want to) solve for y explicitly. Instead, differentiate the whole equation as-is. Every time you differentiate a y-term, attach dydx by the chain rule (since y secretly depends on x), then solve for dydx.

Formal View

If F(x,y)=0 defines y implicitly as a differentiable function of x, then by the chain rule: ∂F∂x+∂F∂y⋅dydx=0, so dydx=−Fx(x,y)Fy(x,y) provided Fy(x,y)≠0.

Worked Examples

Example 1

easy
Find dydx for the circle x2+y2=25 and evaluate it at the point (3,4).

Answer

dydx=−xy; at (3,4): slope =−34

First step

1
Differentiate both sides with respect to x: 2x+2ydydx=0.

Full solution

  1. 2
    Solve for dydx: dydx=−xy.
  2. 3
    At (3,4): dydx=−34.
Whenever a y-term is differentiated, attach dydx by the chain rule. Then collect all dydx terms and solve. The tangent to a circle at (3,4) has slope −3/4.

Example 2

hard
Find dydx for x3+y3=6xy (folium of Descartes).

Example 3

medium
Find dydx for the circle x2+y2=25.

Common Mistakes

  • Differentiating a y-term without attaching dydx - ddx(y3)=3y2dydx, because y depends on x.
  • Forgetting the product rule on mixed xy-terms - ddx(xy)=y+xdydx, not just dydx.
  • Leaving dydx unsolved - after differentiating, collect all dydx terms and solve for it explicitly.

Why This Formula Matters

Many real curves (circles, ellipses, x3+y3=6xy) cannot be solved for y, so implicit differentiation is the only way to get their slopes — and it is the engine behind related rates. It also cements the chain rule: forgetting the dydx tag is the telltale sign a student is still thinking of y as independent. Recognizing it by "Is y tied to x by an equation I can't easily solve for y, and do I need its derivative?" — rather than by familiar numbers — is what lets a student tell it apart from explicit differentiation and chain rule and related rates in a mixed problem set.

Frequently Asked Questions

What is the Implicit Differentiation formula?

Finding dydx when y is defined implicitly by an equation like F(x,y)=0, by differentiating both sides and solving for dydx.

How do you use the Implicit Differentiation formula?

Sometimes you can't (or don't want to) solve for y explicitly. Instead, differentiate the whole equation as-is. Every time you differentiate a y-term, attach dydx by the chain rule (since y secretly depends on x), then solve for dydx.

What do the symbols mean in the Implicit Differentiation formula?

dydx found implicitly. Alternatively, dydx=−FxFy where Fx and Fy are partial derivatives of F(x,y).

Why is the Implicit Differentiation formula important in Math?

Many real curves (circles, ellipses, x3+y3=6xy) cannot be solved for y, so implicit differentiation is the only way to get their slopes — and it is the engine behind related rates. It also cements the chain rule: forgetting the dydx tag is the telltale sign a student is still thinking of y as independent. Recognizing it by "Is y tied to x by an equation I can't easily solve for y, and do I need its derivative?" — rather than by familiar numbers — is what lets a student tell it apart from explicit differentiation and chain rule and related rates in a mixed problem set.

What do students get wrong about Implicit Differentiation?

The procedure for implicit differentiation is the easy part; the trap is differentiating a y-term without attaching dydx. Asking "Is y tied to x by an equation I can't easily solve for y, and do I need its derivative?" first is what keeps a correct-looking calculation from being attached to the wrong concept.

What should I learn before the Implicit Differentiation formula?

Before studying the Implicit Differentiation formula, you should understand: derivative, chain rule, differentiation rules.

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Derivatives Explained: Rules, Interpretation, and Applications →
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