Questions tagged [implicit-differentiation]
For questions on finding and evaluating derivatives when a function is defined implicitly.
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Signed Distance Functions into explicit representations?
I am creating a product which would benefit from conversion of 3d implicit surfaces (also called Signed Distance Functions or F-reps) to explicit boundary representations.
I am creating implicit ...
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Implicit differentiation choice
I was reading Calculus early transcendentals by Howard Anton, in which I encountered an example as follows,
Find the slope of tangents of a sphere $x^2+y^2+z^2=1$ in the direction of $y$ at points $(2/...
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How to find the gradient when it equals to "$\frac00$"?
Find the gradient of the curve $C:ay^3+bx^2y+cxy=1$ where
$$cy+2bxy=0,\quad cx+bx^2+3ay^2=0,$$
if such a point exists.
If we try implicit differentiation, we get
$$3ay^2y'+2bxy+bx^2y'+cy+cxy'=0,$$
...
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Second order partial implicit derivatives
I have some function $F(x, y, z) = 0$, and wish to find the second order cross derivative
$\frac{d^2z}{dxdy}$.
I've easily been able to get the second order derivatives $\frac{d^2z}{dx^2}$ and $\frac{...
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Geometric interpretation of implicit differentiation
It is well known that, given a function $f:\mathbb{R} \to \mathbb{R} $, $f'(x_0)$ can be interpreted as the slope of the tangent line to $f$ in $x_0$. What about curves of the form $F(x, y, c)=0$, ...
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If implicit differentiation yields an undefined expression at a point. Does this mean that derivative is undefined at that point?
Equation $(x-y)^2=0$ implicitly defines two functions: $y=x$. The derivative of each is 1. But the implicit differentiation yields the expression that not defined at $y=x$:
$$(x-y)^2=0$$
$$\frac{d}{dx}...
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Simple Optimization Problem from Economics
An individual has 100 dollars that they have to invest in Asset 1 or 2. The returns from the two assets are $r1\stackrel{d}{=} c + v_1, r2=v2$ for some constant $c$ and independent random variables $...
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Differentiate $x=rcos(\theta)$ with respect to y
So we know: $x=rcos(\theta)$, $y=rsin(\theta)$ and $x^2 + y^2 = r^2$.
I assume it will help to consider $r$ and $\theta$ as functions of $x$ and $y$, but I am not sure how to incorporate this.
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Apply the implicit function theorem of the ratio $c_{t+1}/c_t$ that itselfs depend on the level $c_t$
I'm trying to compute the derivative $\frac{d c_{t+1}/c_t}{d (1+r_{t+1})}$ of this function:
$$\frac{c_{t+1}}{c_{t}} = \left( \beta (1+r_{t+1}) +\gamma \frac{(s-c_{t}+z) ^{-\Sigma }}{c_{t+1}^{-\sigma}}...
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Moving expressions around gives different implicit differentiation
Here's an exercise from Thomas Calculus, I need to do an implicit differentiation:
$$
x^3=\frac{2x-y}{x+3y}
$$
If I enter this in Wolfram Alpha, I get:
$$
y'(x) =-\frac{3 x^4 + 18 x^3 y + 27 x^2 y^2 - ...
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Total derivative of f(x, g(x, y)) and its approximation
I understand the steps to calculate the total derivative of f(x, g(x))
Related: Derivative of $f(x, g(x))$ with respect to $x$
I have three sub-questions related to calculating the total derivative of ...
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Finding the second derivative of $1.51x^2 + y^2 = 1 + 0.71x^2y^2$ to calculate the curvature
Consider the equation
$$1.51x^2 +y^2 = 1 + 0.71x^2y^2.$$
In this question you will calculate the curvature, $\rho$. Evaluate the derivative at the point described — you should get decimal numbers
$$x= ...
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Assume that $\frac{d}{d\theta}\sin\theta = \cos\theta$. Use implicit differentiation to prove that $\frac{d}{d\theta}\cos\theta = - \sin\theta$
Assume that $\frac{d}{d\theta}\sin\theta = \cos\theta$. Use implicit
differentiation to prove that $\frac{d}{d\theta}\cos\theta = - \sin\theta$
Here's my attempt:
$(\sin\theta)^2 + (\cos\theta)^2 = 1$...
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Optimization word problem involving perimeter and area of an arched window
I'm working through an optimization problem in my textbook about maximizing the area of a rectangular window with an arched top.
My question is concerns how to think about a certian variable. I'm torn ...
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Help understanding implicit derivation
I'm currently enrolled in Calculus 1, and everything has been pretty smooth up until these last two sections involving the chain rule and implicit derivation.
After watching multiple YouTube videos, ...
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