Mathematics

Questions tagged [implicit-differentiation]

For questions on finding and evaluating derivatives when a function is defined implicitly.

1350 questions
44
votes
5 answers

Differentiating both sides of an equation

I'm going through the MIT lecture on implicit differentiation, and the first two steps are shown below, taking the derivative of both sides: $$x^2 + y^2 = 1$$ $$\frac{d}{dx} x^2 + \frac{d}{dx} y^2 = \frac{d}{dx} 1$$ $$2x + \frac{d}{dx}y^2 = 0$$ That…
Jon
  • 583
19
votes
4 answers

Can someone give me a deeper understanding of implicit differentiation?

I'm doing calculus and I want to be an engineer so I would like to understand the essence of the logic of implicit differentials rather than just memorizing the algorithm. Yes, I could probably memorize it and get a 100% on a test, but it means…
Klik
  • 951
15
votes
4 answers

Why does $\frac{dq}{dt}$ not depend on $q$? Why does the calculus of variations work?

The Euler–Lagrange equations for a bob attached to a spring are $${d\over dt}\left({\partial L\over\partial v}\right)=\left({\partial L\over\partial x}\right)$$ But is $v$ a function of $x$ ? Normal thinking says that $x$ is a function of $t$ and…
Shashaank
  • 933
14
votes
1 answer

What's the arc length of an implicit function?

While an explicit function $y(x)$'s arc length $s$ is easily obtained as $$s = \int \sqrt{1+|y'(x)|^2}\,dx,$$ is there any formula for implicit functions given by $f(x,y) = 0$? One can use the implicit differentiation $y'(x) = -\frac{\partial_y…
Tobias Kienzler
  • 6,824
13
votes
1 answer

How to show that the fabius function is nowhere analytic?

Consider the fabius function https://en.m.wikipedia.org/wiki/Fabius_function https://people.math.osu.edu/edgar.2/selfdiff/ How does one show that this function is nowhere analytic ? Probably related , Maybe even a step in the answer : how to…
mick
  • 18,413
12
votes
3 answers

Finding the turning points of $f(x)=\left(x-a+\frac1{ax}\right)^a-\left(\frac1x-\frac1a+ax\right)^x$

I've just come across this function when playing with the Desmos graphing calculator and it seems that it has turning points for many values of $a$. So I pose the following problem: Given $a \in \mathbb{R}-\{0\}$, find $x$ such that…
TheSimpliFire
  • 28,628
  • 10
  • 66
  • 140
12
votes
1 answer

Smooth sawtooth wave $y(x)=\cos(x-\cos(x-\cos(x-\dots)))$

Consider an infinite recursive function $$y(x)=\cos(x-\cos(x-\cos(x-\dots)))$$ $$y=\cos(x-y)$$ Plotting the function $y(x)$ implicitly we get a smooth sawtooth-like wave: Was this function studied before? For example, its derivative, Fourier series…
Yuriy S
  • 32,998
12
votes
5 answers

How is implicit differentiation formally defined?

I get that differentiation is an operation used on a function, so if a function is defined $x\mapsto x^2$, the derivative is $$ (x\mapsto x^2)' = x \mapsto \lim_{h\to 0} \frac{x^2+2xh+h^2-x^2}{h} = x\mapsto 2x. $$ But how can you extend the…
Frank Vel
  • 5,537
11
votes
1 answer

Is the differential forms perspective on $dx$ incompatible with the technique of implicit differentiation?

Suppose $$x^2 + y^2 = 5^2.$$ We're trying to find $dy/dx$ at $(3,4).$ Applying $d$ to both sides: $$2x dx + 2y dy = 0$$ Or in other words: $$2x dx + 2y dy = 0dx + 0dy$$ Since the covectors $dx_p$ and $dy_p$ form a basis for the cotangent space at…
goblin GONE
  • 69,755
11
votes
6 answers

When to write "$dx$" in differentiation

I'm taking differential equations right now, and the lack of fundamental knowledge in calculus is kicking my butt. In class, my professor has done several implicit differentiations. I realize that when taking the derivative with respect to "$x$," I…
Sam
  • 113
10
votes
5 answers

Why does implicit differentiation work on non-functions?

I've been reading Keisler's Calculus book, and there's an example where he does implicit differentiation on the equation: $$x^2+y^2=1$$ which yields:$$\frac{dy}{dx}=-\frac{x}{y}$$ I understand the technique of implicit differentiation, and I…
user17137
10
votes
1 answer

An expression for computing second order partial derivatives of an implicitely defined function

Let $\Phi(x,y)=0$ be an implicit function s.t. $\Phi:\mathbb{R}^n\times \mathbb{R}^k\rightarrow \mathbb{R}^n$ and $\det\left(\frac{\partial \Phi}{\partial x}(x_0,y_0)\right)\neq 0$ . This means that locally at $(x_0,y_0)$ we can express $x_i$ as…
Dmitry
  • 1,345
10
votes
4 answers

Implicit differentiation involving a sliding ladder

A $5$-foot long ladder is resting on a wall, so that the top of the ladder is 4 feet above the ground and the bottom of the ladder is $3$ feet from the wall. At some time, the ladder is slipping so that the top of the ladder falls at a constant rate…
Data
  • 930
10
votes
3 answers

Extraneous and Missing Solution Confusion

I came to a question in James Stewart's "Calculus Early Transcendentals" about implicit differentiation saying that: Find all points on the curve $x^2y^2+xy=2$ where the slope of the tangent line is $-1$ After implicit differentiation I came to…
Zyad Yasser
  • 546
10
votes
3 answers

What does the denominator in the second derivative mean?

This occurred to me a few days ago. We know that the derivative of a function $y=f(x)$ is $\frac{dy}{dx}$. This is because it represents how $y$ changes with $x$, which is the rate of change of $y$, or more specifically, the gradient of a function.…
AkThao
  • 279
1
2 3
…
89 90