Implicit Differentiation

Reading

• Sections 3.8

Practice problems

• Section 3.8: 3, 5, 7, 9, 11, 31, 33, 39
• To turn in (together with 3.9): 3.8 6, 10, 38

Notes

Implicit Differentiation

Implicit differentiation is used when we have a relation between two variables \(x\), \(y\), which would in theory allow us to write \(y\) as a function of \(x\), but when actually doing so might be hard to do.

Example:

Let us look at a case that we can solve in two ways.

• Imagine we have the equation: \(x^2 + y^2 = 1\).
• Then \(y\) can be thought of as a function of \(x\).
• We could in fact solve it, and get \(y = \sqrt{1-x^2}\).
• Then we can compute the derivative by chain rule: \(\frac{dy}{dx} = \frac{-x}{\sqrt{1-x^2}}\).
• Alternatively, we can differentiate the formula \(x^2+y^2 = 1\) treating \(y=y(x)\) as a function of \(x\).
• This gives us: \(2x + 2y \frac{dy}{dx} = 0\)
• Solving for \(\frac{dy}{dx}\) gives us: \(\frac{dy}{dx} = - \frac{x}{y}\)
• Since \(y=\sqrt{1-x^2}\), these two formulas are the same.

Implicit Differentiation

Instead of a function \(y=y(x)\), we are given a formula \(F(x, y) = c\) that relates \(x\) and \(y\).

We differentiate that formula with respect to \(x\), treating \(y\) as a function of \(x\).

Then we solve the resulting equation for \(\frac{dy}{dx}\). The result will contain both \(x\) and \(y\).

Key property: The chain rule tells us that if \(y\) is a function of \(x\), then: \[\frac{d}{dx}f(y) = f'(y) \frac{dy}{dx}\]

Example: Consider the relation \(y^3+xy^2=x^2+1\). The point \((1, 1)\) belongs to this curve. We want to find the equation of the tangent line to this graph at that point.

The equation would be: \[y-1 = \left.\frac{dy}{dx}\right|_{(x,y)=(1,1)}(x-1)\] We use implicit differentiation to compute that derivative. We have: \[3y^2\frac{dy}{dx} + y^2 + 2xy\frac{dy}{dx} = 2x\] Solving for \(\frac{dy}{dx}\): \[\frac{dy}{dx} = \frac{2x-y^2}{3y^2+2xy}\] Plugging in \((x, y) = (1, 1)\): \[\frac{dy}{dx} = \frac{1}{5}\] So tangent line equation becomes: \[y-1 = \frac{1}{5}(x-1)\]