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Calculus IDerivatives

Implicit differentiation, one side at a time.

Find slopes even when x and y are tangled together. Learn the method, explore tangent lines, then solve every class question with guided working.

Differentiate a relation · Step 1 of 7

Find a slope without isolating y

−4−3−2−101234−2−1012
An explicit formula has y{y} alone, such as y=x2{y} = {x^2}. An implicit relation mixes the variables. The circle x2+y2=4{x^2+y^2} = {4} is not one function of x: most x-values have two y-values. But near a point away from its left and right tips, we can follow one smooth branch as a function of x.
Try it: Move the point around the circle. The blue dashed tangent follows it. At the left and right tips the tangent is vertical, so there is no finite slope.
Position around the circle60°
y=4−x2{\displaystyle y} = {\displaystyle \sqrt{4-x^2}}
y=−4−x2{\displaystyle y} = {\displaystyle -\sqrt{4-x^2}}

The first formula is the upper half; the second is the lower half. Implicit differentiation handles both using the original equation.

Check yourself
Is the whole circle a single function of x?

Need help keeping track of every y term?

Bring your practice questions. We work through them together until every type feels routine.

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