Implicit differentiation, what's going on here? | Chapter 6, Essence of calculus
Implicit differentiation can feel weird, but what's going on makes much more sense once you view each side of the equation as a two-variable function, f(x, y).
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Let me share with you something I found particularly weird when I was a student first learning calculus. Let’s say you have a circle with radius 5 centered at the origin of the xy-coordinate plane, which is defined using the equation x2 + y2 = 52. That is, all points on this circle are a distance 5 from the origin, as encapsulated by the pythagorean theorem with the sum of the squares of the legs of this triangle equalling the square of the hypotenuse, 52. And suppose you want to find the slope of a tangent line to this circle, maybe at the point (x, y) = (3, 4). Now, if you’re savvy with geometry, you might already know that this tangent line is perpendicular to the radius line touching that point. But let’s say you don’t already know that, or that you want a technique that genera...
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Let me share with you something I found particularly weird when I was a student first learning calculus. Let’s say you have a circle with radius 5 centered at the origin of the xy-coordinate plane, which is defined using the equation x2 + y2 = 52. That is, all points on this circle are a distance 5 from the origin, as encapsulated by the pythagorean theorem with the sum of the squares of the legs of this triangle equalling the square of the hypotenuse, 52. And suppose you want to find the slope of a tangent line to this circle, maybe at the point (x, y) = (3, 4). Now, if you’re savvy with geometry, you might already know that this tangent line is perpendicular to the radius line touching that point. But let’s say you don’t already know that, or that you want a technique that genera...
