Implicit differentiation, what's going on here? | Chapter 6, Essence of calculus
Why Mathematics Stack Exchange recommends it
Mathematics Stack Exchange users link it when someone needs an intuitive picture of what implicit differentiation is doing, and askers also cite it when puzzling over its treatment of differentials.
Recommended by 3 people across 3 different months, 2017 to 2021.
What it is
Chapter 6 of 3Blue1Brown's Essence of Calculus series, narrated and animated by Grant Sanderson. It builds intuition for implicit differentiation using a circle and a sliding ladder, treating equations as constraints on small changes dx and dy, and finishes by deriving the derivative of ln(x).
Cited in 1 Wikipedia article, including Implicit function.
What people said
Strong rec Typo · 2021-11 · Mathematics Stack Exchange · on “What precisely is "implicit differentiation" and what results does it use?”
Excellent video on the intuition of implicit differentiation by 3Blue1Brown [this video] . Might be worth checking out
Recommends Yannik · 2017-10 · Mathematics Stack Exchange · on “Can we differentiate $x^y =-1$ ? Why?”
The relation $y\log x =\log 4$ reveals that $$y' = -\frac{y}{x\log x} = -\frac{\log 4 / \log x}{x\log x} = -\frac{\log 4}{x\log^2 x}$$ which is what to expect since $y(x) = \log 4 /\log x$. Youtuber 3Blue1Brown has a nice video [this video] covering the intuition behind this.
Recommends Hazzaldo · 2019-03 · Mathematics Stack Exchange · on “Calculus - Implicit Curve - rearranging derivative equation”
I'm new to this community, so please bear with me if I make a mistake following rules and conventions, and thanks in advance. I'm currently learning - following along from this video [this video] ("Implicit differentiation, what's going on here? | Essence of calculus"). At 2:42 in the video timeline, the teacher mentions that the implicit curve equation $x^2+y^2=5^2$ , which is then resolved into its derivative form: $2x\,\mathrm{d}x+2y\,\mathrm{d}y=0$ can then be rearranged to the form expression: $\frac{\mathrm dy}{\mathrm dx} = -\frac{x}{y}$ . I wanted to understand how this rearrangement happened? I tried to explore this equation rearrangement: $2x\,\mathrm{d}x+2y\,\mathrm{d}y=0$ to $\frac{\m
Cites zazke · 2018-05 · Mathematics Stack Exchange · on “If dx is a variable, then how can we divide by it since we know it can be zero.”
I thought I knew that we use differentials just because they made calculus intuitive and there was math rigor behind all of these. But when I saw this 3B1B video about the topic [this video] this problem came out. For example, deriving implicitly: $$ x^2+y^2 = 5 $$ we get $$ 2x\mathrm{d}x +2y\mathrm{d}y = 0 $$ $$ 2xdx+2y\frac{dy}{dx}dx=0 $$ and then to find the slope of the graph we solve for dy/dx $$ \frac{dy}{dx}=-\frac{x}{y} $$ Does't that imply that $dx \neq 0$ ? And how would it be in the general case whe differentiating implicitly