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 $y'(x) = -\frac{\partial_y f}{\partial_x f}$ to obtain $$s = \int\sqrt{1 + |\partial_y f / \partial_x f|^2}\,dx,$$ but that still requires (locally) […]

I’m totally lost on the following question: I need to find the equation of the line tangent to the ellipse $$\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$$ at the given point $(m, n)$.

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 have to write “$\frac{dy}{dx}$” whenever I differentiate “$y$” and nothing when it’s “$x$” since it will […]

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 nothing unless I understand it and can acquire a […]

In high school, we learned to reason like so: $$(*) \qquad \frac{d}{dx}(x^2+x) = \frac{d}{dx}(x^2)+\frac{d}{dx}(x) = 2x+1$$ Now that I know more, I can “reanalyze” this chain of reasoning using ideas that I have more faith in, like “function” and “lambda abstraction.” We begin by defining $\nabla (f)$ as the derivative of $f$. Then the above […]

I have to use implicit differentiation to find $\frac{dy}{dx}$ given: $$x^2 \cos(y) + \sin(2y) = xy$$ I don’t even know where to begin, I missed the class where we went over implicit differentiation, and because of that, I am completely stuck. Thank you everyone. Edit: I don’t know how to make the equation look all […]

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 $v$ is a function of $t$, but it is not necessary that $v$ be a function of $x$. Mathematically, however, […]

I want to differentiate $x^2 + y^2=1$ with respect to $x$. The answer is $2x +2yy’ = 0$. Can some explain what is implicit differentiation and from where did $y’$ appear ? I can understand that $2x +2yy’ = 0$ is a partial derivative but then it becomes multi calc not single. This is in […]

we are given the function $F: \mathbb R^3 \to \mathbb R^2$, $F(x,y,z)=\begin{pmatrix} x+yz-z^3-1 \\ x^3-xz+y^3\end{pmatrix}$ Show that around $(1,-1,0)$ we can represent $x$ and $y$ as functions of $z$, and find $\frac{dx}{dz}$ and $\frac{dy}{dz}$ What I did: The differential of $F$ is: $\begin{pmatrix} 1 & z &y-3z^2\\3x^2-z & 3y^2 &-x\end{pmatrix}$, insert $x=1,y=-1,z=0$ to get: $\begin{pmatrix} […]

I need to know the equation for a line tangent to a circle and through a point outside the circle. I have found a number of solutions which involve specific numbers for the circles equation and the point outside but I need a specific solution, i.e., I need an equation which gives me the $m$ […]

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