# Tagged Questions

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

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### Numerical Methods For Implicit Differentiation

I am implementing a Genetic Programming algorithm that uses a special cost function to perform Symbolic Regression (see this paper) (essentially curve fitting, but instead of finding parameters/...
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### Asymptote of non-algebraic implicit function

The function f is implicitly given by $g(x,y)=e^{xy}-x-y=0$. I want to know about the asymptotes towards $+\infty$. I could deduce that there is only one intersect with the x-axis at (1|0) and by the ...
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### Implicit differentiation of a two variables function

A function $z=z(x,y)$ is given implicitly by the function $$f\left(\frac{x}{y},\frac{z}{x^{\lambda}}\right)=0$$ where $\lambda\in\mathbb{R},\lambda\neq0$. I have to show that if $f(u,v)$ is ...
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### find ${dy}/{dx}$ if $x^y + y^x = 1$

Find ${dy}/{dx}$ if $x^y + y^x = 1$. I have no idea how to approach this problem. Can somebody please explain this to me?
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### Finding equation of tangent lines to hyperbola $xy=1$

I am working through some calculus problems (this is in a section on implicit differentiation) and this one is giving me trouble. I am trying to find the equations of the tangent lines to the ...
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### For $f_{a,b}(x)=2^{x+a-b}-abx+4$, show that for $x_1,x_2$ close enough to $2,3$ there are $a,b$ such that $f_{a,b}(x_1)=f_{a,b}(x_2)=0$

Let $f_{a,b}(x)=2^{x+a-b}-abx+4$. Show that for $x_1$ close enough to $2$ and $x_2$ close enough to $3$ there are $a,b$ such that $f_{a,b}(x_1)=f_{a,b}(x_2)=0$. Hint: $f_{2,2}(2)=f_{2,2}(3)=0$. It ...
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### Which derivative is correct?

Consider the following expressions: $$C_{i}=\sum_{j=1}^{N_i}v_{j}, \quad v_j \in \mathbb{R}, \quad N_i \in \mathbb{N}$$ $$x_{i}=\frac{C_{i}}{N_i}$$ I want to obtain an ...
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### Implicit differentiation of Product of Two Functions of $y$

I am asked to find the derivative of $$e^y\cos^2y$$ with respect to $x$. I think it is $$y'e^y\cdot2y'\cos y \sin y$$ Since there is no $x$ and $y$ term, such as $xy$, the product rule does not apply(...
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### Differential equations with inverse trigonometrical functions

$(x^2+y^2)^(1/2)$=$e^(asin(y/(x^2+y^2)^1/2))$ Prove that $\frac{d^2 y}{dx^2}$=$2((x^2+y^2))/(x-y)^3$, x>0 I started with taking the natural log of the given equation and differentiating it, I ended ...
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### How to find the slope of curves at origin if the derivative becomes indeterminate

What's the general method to find the slope of a curve at the origin if the derivative at the origin becomes indeterminate. For Eg-- What is the slope of the curve $x^3 + y^3= 3axy$ at origin and how ...
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### Relation between chain rule and implicit differentiation derivation in multi variable calculus

So my question is on the derivation of the implicit differentiation (taken from here). The general chain rule, from here, it says that if we have a function $z$ of $n$ variables, $x_1, x_2,\ldots,x_n$...
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### Find the tangent lines to the graph of $x^2+4y^2 = 36$ that go through the point $P=(12,3)$

I was solving a few problems from a textbook and I came across this one: Find the tangent lines to the graph of $x^2+4y^2 = 36$ that go through the point $P=(12,3)$ I could find the tangency ...
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### Implicit derivation to find $\partial x/\partial v$?

I saw this question: $$\begin{cases} x^2+y^2=u \\ x\sin y+y=v\end{cases}$$ What is the $\partial x/\partial v$? I think it should be $1/\sin(y)$ because $\partial v/\partial x=\sin y$, but the ...
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### Implicitly finding the derivative of $f^{-1}(x)$ given $f(x)$

Can we find the derivative of the inverse of a function implicitly by finding the derivative of the original function? For example lets say I have $f(x) = e^x$ and I want to find the derivative of ...
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### Determine the point on the curve $a ^ 2 x ^ 2 + y ^ 2 = a ^ 2$ in the first quadrant such that the area of ​the triangle by tangent

Problem: Let the $a$ arbitrary . Determine the point on the curve $a ^ 2 x ^ 2 + y ^ 2 = a ^ 2$ in the first quadrant such that the area of ​​the triangle by tangent the curve drawn at this point ...
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### Prove that $F(x)=\alpha(x)f(x)$ is differentiable and compute the derivative

For an assignment, I have to solve this problem, but I just can't figure out how to continue. I already figured it out for the case $F(x)=f(x)+g(x)$ but this one I just cannot figure out. Would be ...