# Implicit Differentiation

## Definition

Implicit Differentiation is an approach to taking derivatives that uses the chain rule to avoid solving explicitly for one of the variables.

For example, if $$y + 3x + 8 = 0$$, we could solve for $$y$$ and then differentiate:

\begin{aligned} y &= -3x - 8 \\ \frac{dy}{dx} &= -3 \end{aligned}

However, we could also simply take the derivative of each term with respect to $x$ in place:

\begin{aligned} y + 3x + 8 &= 0 \\ \frac{dy}{dx} + 3 + 0 &=0 \\ \frac{dy}{dx} &= -3 \end{aligned}

The second approach is known as implicit differentiation.

## Technique

### Given $x^2 + x + y^2 = 15$ what is $\frac{dy}{dx}$ at the point $( 2, 3)$?

Since $x^2 + x + y^2 = 15$, differentiating, we have:

\begin{aligned} 2x+1+2y\left(\frac{dy}{dx}\right)&=0 \\ \frac{dy}{dx}&=\frac{2x+1}{2y} \end{aligned}

Thus $\frac{dy}{dx}$ at the point $( 2, 3)$ is $\frac{2(2)+1}{2(3)}=\frac{5}{6}$. $_\square$

### If $y^6 - e^{xy} = x$ what is $\frac{dy}{dx}$?

Taking the derivative of every term with respect to $x$ gives us:

\begin{aligned} 6y^5\left( \frac{dy}{dx} \right) - e^{xy} \left(\frac{d}{dx} (xy)\right)&=1 \\ 6y^5\left( \frac{dy}{dx} \right) - e^{xy} \left( y + x\frac{dy}{dx} \right)&=1 \end{aligned}

Now, we can isolate all of the $\frac{dy}{dx}$ on the left:

\begin{aligned} 6y^5\left( \frac{dy}{dx} \right) - xe^{xy}\left(\frac{dy}{dx}\right) &=1 + ye^{xy} \\ \left( \frac{dy}{dx} \right) \left(6y^5 - xe^{xy} \right)&=1 + ye^{xy} \\ \frac{dy}{dx} &= \frac{ 1 + ye^{xy} }{ 6y^5 - xe^{xy} } _\square \end{aligned}

Note by Arron Kau
6 years, 11 months ago

This discussion board is a place to discuss our Daily Challenges and the math and science related to those challenges. Explanations are more than just a solution — they should explain the steps and thinking strategies that you used to obtain the solution. Comments should further the discussion of math and science.

When posting on Brilliant:

• Use the emojis to react to an explanation, whether you're congratulating a job well done , or just really confused .
• Ask specific questions about the challenge or the steps in somebody's explanation. Well-posed questions can add a lot to the discussion, but posting "I don't understand!" doesn't help anyone.
• Try to contribute something new to the discussion, whether it is an extension, generalization or other idea related to the challenge.

MarkdownAppears as
*italics* or _italics_ italics
**bold** or __bold__ bold

- bulleted
- list

• bulleted
• list

1. numbered
2. list

1. numbered
2. list
Note: you must add a full line of space before and after lists for them to show up correctly
paragraph 1

paragraph 2

paragraph 1

paragraph 2

> This is a quote
This is a quote
# I indented these lines
# 4 spaces, and now they show
# up as a code block.

print "hello world"
# I indented these lines
# 4 spaces, and now they show
# up as a code block.

print "hello world"
MathAppears as
Remember to wrap math in $$...$$ or $...$ to ensure proper formatting.
2 \times 3 $2 \times 3$
2^{34} $2^{34}$
a_{i-1} $a_{i-1}$
\frac{2}{3} $\frac{2}{3}$
\sqrt{2} $\sqrt{2}$
\sum_{i=1}^3 $\sum_{i=1}^3$
\sin \theta $\sin \theta$
\boxed{123} $\boxed{123}$

Sort by:

nice

- 6 years, 9 months ago