Why Find a Second Derivative Implicitly?
Some curves are much easier to describe with an equation involving both and than by solving explicitly for . A circle such as is a simple example.
Implicit differentiation lets you find the first derivative without solving for . If a problem asks for the second derivative, however, you must differentiate your first-derivative equation one more time.
The important idea is that is itself a function of . When you differentiate a term containing again, the chain rule appears automatically.
The Key Formula
Suppose an equation defines implicitly. After the first differentiation, you obtain an expression for . Differentiate that entire equation with respect to again:
A particularly important result occurs when the first derivative appears in the equation. Because is a function of , its derivative is:
That is why you will often see appear when finding the second derivative implicitly. You are differentiating using the chain rule.
A Step-by-Step Method
- Differentiate the original equation once with respect to , treating as a function of .
- Solve the first-derivative equation for , whenever practical. This makes the second differentiation much easier.
- Differentiate the first-derivative equation again. Remember that must be differentiated with the chain rule.
- Solve for . The second derivative may appear on both sides of the equation.
- Substitute the coordinates of the requested point, along with the corresponding value of , if needed.
Worked Examples
Find for the curve . Then find its value at the point .
Find for .
The curve passes through . Find the value of at that point.
Suppose an implicit equation produces the first derivative
To find the second derivative, differentiate both sides. The term involving requires the chain rule.
Why the Chain Rule Matters the Second Time
The first implicit differentiation is usually straightforward: every time you differentiate a term involving , a factor of appears.
The second differentiation is where students often lose track of what is happening. For example, if your first derivative contains , then differentiating that expression gives:
The extra factor of is not optional. It comes from the chain rule because is not an independent constant; it depends on .
Common Mistakes
If your first derivative contains , do not treat it like a constant. Its derivative is . This is the single most important idea in second-order implicit differentiation.
After differentiating twice, your equation may contain both and . If the problem asks for a numerical value, you usually need the first derivative at the point before you can finish evaluating the second derivative.
You can sometimes solve an equation for , but doing so may create unnecessary square roots and multiple branches. Implicit differentiation is specifically designed to avoid that work.
Finding is only half the job when the question asks for . Read the requested derivative carefully before stopping.
Practice Problems
Find for the curve .
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Find for at the point .
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The curve passes through . Find at that point.
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For , find in terms of , , and .
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