Скачать или смотреть But where does dy/dx come from? Implicit differentiation is really just the chain rule!

Implicit differentiation can be easy if you see it as just another application of the chain rule!

It happened again this semester in my Calculus 1 class: we were solving implicit differentiation problems and a student asked me "where does the dy/dx come from?". This is a super common question because implicit differentiation can be tricky to understand, but once you get that implicit differentiation is really just the chain rule, it's no harder than any other chain rule problem.

So why does implicit differentiation work, and where does the dy/dx come from? We start with a basic chain rule example: the derivative of (x^2+3x)^3. To take the derivative, we differentiate with respect to the interior function to get 3(x^2+3x)^2 then the chain rule tells us to tack on the derivative of the interior function, which is (2x+3).

So what if I told you that y is some unknown function of x, and I'd like you to take the x-derivative of y^3? To find d/dx(y^3), you do exactly the same thing as before: take the derivative with respect to the interior function y, giving 3y^2, then tack on the derivative of y, which is dy/dx or y'(x) if you prefer the prime notation. THAT'S where the dy/dx comes from: it's just the chain rule tacking on the derivative of the interior function.

So why do we care about implicit differentiation? We give one more example where we're asked to find the tangent line to an implicit curve. To isolate dy/dx (the slope of the curve), we differentiate both sides with respect to x, using implicit differentiation (i.e. the chain rule) to generate factors of dy/dx. We isolate the dy/dx to get the slope of the tangent line, apply the point slope formula, and we're done!

#calculus #implicitdifferentiation #chainrule