How does implicit differentiation work?

Question

alienceenvedsf0 · Accepted Answer

Implicit differentiation is a way of differentiating when you have a function in terms of both x and y. For example:

x^{2} + y^{2} = 16

This is the formula for a circle with a centre at (0,0) and a radius of 4
So using normal differentiation rules

x^{2}

and 16 are differentiable if we are differentiating with respect to x

\frac{d}{d x} (x^{2}) + \frac{d}{d x} (y^{2}) = \frac{d}{d x} (16)

2 x + \frac{d}{d x} (y^{2}) = 0

To find

\frac{d}{d x} (y^{2})

we use the chain rule:

\frac{d}{d x} = \frac{d}{d y} \cdot \frac{d y}{d x}

\frac{d}{d y} (y^{2}) = 2 y \cdot \frac{d y}{d x}

2 x + 2 y \cdot \frac{d y}{d x} = 0

Rearrange for

\frac{d y}{d x}

\frac{d y}{d x} = \frac{- 2 x}{2 y}

\frac{d y}{d x} = - \frac{x}{y}

So essentially to use implicit differentiation you treat y the same as an x and when you differentiate it you multiply be

\frac{d y}{d x}

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