What Implicit Differentiation Is For
Most of the functions you've differentiated so far have been written explicitly: . In that form, y is already isolated, so finding is usually straightforward.
But many important curves are not written that way. Consider the equation . Here, x and y are mixed together, so solving for y first may be difficult, inconvenient, or even produce multiple branches.
Implicit differentiation lets you find directly from the original equation. You differentiate both sides with respect to x, treating y as a function of x.
The Key Idea: y Is Really a Function of x
When you see a term such as , it may look as though you're simply differentiating a power of y. But because y depends on x, the chain rule is involved.
For example:
The extra factor appears because y is changing as x changes. This is the central idea behind implicit differentiation.
The Basic Rules
When differentiating an equation implicitly with respect to x, keep these rules in mind:
- Differentiate every term on both sides.
- Treat y as a function of x. Whenever you differentiate a term involving y, the chain rule may introduce .
- Terms involving only x are differentiated normally.
- Constants differentiate to zero.
- After differentiating, collect all terms containing on one side and solve for .
A Step-by-Step Method
- Differentiate both sides with respect to x. Do not solve for y first.
- Apply the chain rule to every term involving y. This is why appears.
- Keep visible while you simplify. It is usually easiest to collect those terms together.
- Factor out .
- Solve for .
- If a point is provided, substitute its coordinates only after obtaining the derivative formula, unless doing so earlier clearly simplifies the algebra.
Worked Examples
Find for the circle .
Find the slope of the curve at the point .
Find for .
Find for .
Why the Chain Rule Appears
The notation can seem strange at first. Why does differentiating produce instead of simply ?
The reason is that y is not an independent constant when we differentiate with respect to x. It is a function of x: . So the derivative of is really the derivative of a composite function.
Once you understand this, the appearance of stops being a special trick. It is simply the chain rule.
Common Mistakes
The most common error is treating as though y were just another constant with respect to x. Because y depends on x, differentiating a power of y requires the chain rule.
When a term contains both x and y, such as , you generally need the product rule and the chain rule. Do not differentiate the term as though only one factor changes.
You can sometimes solve an equation explicitly, but that is often unnecessary and may create square roots, plus-or-minus branches, or complicated algebra. Implicit differentiation is designed to avoid that step.
For most AP Calculus problems, it is cleaner to derive a general expression for first and then substitute the given coordinates.
Practice Problems
Find for .
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Find for .
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Find the slope of the curve at the point .
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Find for .
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