Implicit differentiation
Differentiating both sides of an equation with respect to x to find dxdy for relations not written as y=f(x), using the product and chain rules, and finding horizontal and vertical tangents and normals from the resulting derivative.
Implicit Differentiation
Implicit differentiation is when we differentiate both sides of an equation. It is helpful when we have an equation that cannot be simplified to y=f(x).
For example:
Differentiating both sides with respect to x and using the product rule:
By the chain rule, we know that (y2)′=2y⋅dxdy:
Now collecting dxdy terms:
So
Implicit Derivative: Horizontal and vertical tangents & normals
In problems involving implicit derivatives, you may be asked to solve for points where the tangent to the curve is horizontal or vertical. A horizontal tangent means dxdy=0, and a vertical tangent occurs in the case where dxdy=denominatornumerator and the denominator equals zero.
If the question asks for vertical / horizontal normals, just recall that a vertical normal means a horizontal tangent, and vice-versa.
Nice work completing Implicit differentiation, here's a quick recap of what we covered:
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