Calculus How To

Finding the Second Derivative Implicitly

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Like the “usual” way of finding second derivatives, finding the second derivative implicitly involves two steps: implicitly differentiating twice.

The key to finding the second derivative implicitly requires a good understanding of the chain rule. For example, to take the derivative of an expression like 4y2 with respect to x, you have an inside function and an outside function:

  1. Take the derivative with respect to y,
  2. Multiply by y′.

That results in 8yy′.

Finding the Second Derivative Implicitly: Example

Example question: Find the second derivative implicitly of x2 + 4y2 = 1.

Part One: Finding the First Derivative Implicitly

Step 1: Take the derivative(s) of the left hand side. We have two parts to differentiate: x2 and 4y2:

  • x2 (using the power rule) =
    2x
  • 4y2:
    1. Take the derivative with respect to y,
    2. Multiply by y′.

    Solution: 8yy′.

Combining the answers (because of the Sum Rule) we get:
2x + 8yy

Step 2: Take the derivative of the right hand side. The derivative of a constant is 0.

Step 3: Place your answers from the left hand side (Step 1) and right hand side (Step 2) back together:

2x + 8yy′ = 0

This is the first implicit derivative.

You may be wondering why I split this up into three parts. After all, finding the derivative of 25 is something you can probably do in an instant. That’s because some implicit derivatives are more complicated (e.g. 3xy2 – 2x = 4y) and breaking the problems up into small chunks can make it easier to keep track of the parts.


Part Two: Finding the Second Derivative Implicitly

You could jump right in at this point and re-run the steps above to get the second derivative. However, you’ll end up with yy′′ in the solution, which means you’ll have to backtrack and solve the earlier equation for y′.
Rather than do that, I like the approach given on MIT’s Open Courseware site [1]: It’s better to streamline and solve for y′ first.

Step 3: Solve for y′:


  • 2x + 8yy′ = 0
  • 8yy′ = – 2x
  • y′ = – 2x/8y
  • y′ = – x/4y

Step 4: Differentiate both sides again. The right hand side of this particular equation is solved with the quotient rule:

finding the second derivative implicitly step 4

Step 5: Substitute –x/4y for y(Step 3):
substituting y in implicit equation

Step 6: Substitute in the original equation x2 + 4y2 = 1. (Note: A popular online calculator skipped this step!):

Solution: y′′ = -(1 / 16y3).

References

[1] MIT: Retrieved April 17, 2021 from: https://ocw.mit.edu/courses/mathematics/18-01sc-single-variable-calculus-fall-2010/1.-differentiation/part-b-implicit-differentiation-and-inverse-functions/session-13-implicit-differentiation/MIT18_01SCF10_ex13sol.pdf (CC BY-NC-SA 4.0).

CITE THIS AS:
Stephanie Glen. "Finding the Second Derivative Implicitly" From CalculusHowTo.com: Calculus for the rest of us! https://www.calculushowto.com/finding-the-second-derivative-implicitly/
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