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 4*y*^{2} with respect to *x*, you have an **inside function** and an **outside function**:

- Take the derivative with respect to
*y*, - Multiply by
*y*′.

That results in 8*yy*′.

## Finding the Second Derivative Implicitly: Example

**Example question**: Find the second derivative implicitly of x^{2} + 4y^{2} = 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: *x*^{2} and 4*y*^{2}:

(using the power rule) =*x*^{2}

2*x***4***y*^{2}:- Take the derivative with respect to
*y*, - Multiply by
*y*′.

Solution: 8

*yy*′.- Take the derivative with respect to

Combining the answers (because of the Sum Rule) we get:

2*x* + 8*yy*′

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**:

2*x* + 8*yy*′ = 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. 3*xy*^{2} – 2*x* = 4*y*) 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′:**

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

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

Step 5: **Substitute – x/4y for y′ **(Step 3):

Step 6: **Substitute in the original equation **x^{2} + 4y^{2} = 1. (Note: A popular online calculator skipped this step!):

**Solution**: y′′ = -(1 / 16*y*^{3}).

## 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/

**Need help with a homework or test question? **With Chegg Study, you can get step-by-step solutions to your questions from an expert in the field. Your first 30 minutes with a Chegg tutor is free!