# Questions tagged [implicit-differentiation]

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

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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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### 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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### Can you square root both sides of an implicit equation to get the derivative? [closed]

Suppose you have $x^2=(4x^2y^3 + 1)^2$, can you square root both sides of the equation to make it simpler?
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### If $f(x)=\frac{2}{\sqrt{3}}\int_{0}^{\sqrt{3}}f\left(\frac{\lambda^2x}{3}\right)d\lambda,x>0$ and $f(1)=\sqrt3$ and if $f(\alpha)=6$ . find $\alpha$

Let $f(x)$ be a differentiable function satisfying $$f(x)=\frac{2}{\sqrt{3}}\int_{0}^{\sqrt{3}}f\left(\frac{\lambda^2x}{3}\right)d\lambda,x>0$$ and $f(1)=\sqrt3$ . If $y=f(x)$ passes through the ...
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### implicit differential equation, y is a solution [closed]

Assume that the function $y$ is defined implicitly as a function of $x$ by the given equation. Use the technique of implicit differentiation to find a differential equation for which $y$ is a solution....
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I have that $x^{*}(w,z)$ and $y^{*}(w,z)$ is the implicit solution to a the system $F(x^{*}, y^{*},w) = 0$ and $H(x^{*}, y^{*},z) = 0$. Using the implicit function theorem, I can compute $\frac{\... 2 votes 1 answer 120 views ### Why do we call$y$a function of$x$in implicit differentiation? When we have something like$y = 2x$we understand$y$to be the value of the function$f$at each point$x$where$f(x) = 2x$, to reiterate,$y$is not a function but merely a label for the output of ... • 1,057 0 votes 0 answers 60 views ### Mapping the derivative of an implicit function on a 2D plane I'm in the middle of my first calculus class, a week ago we covered how to find the derivative of implicit functions, and I'm still thinking about it. I completely understand what I am supposed to ... 1 vote 2 answers 61 views ### Stuck on implicit differentiation exercise; inquiring about how to approach it On exercise 8 on section 3.6 of Stewart’s Second Edition Single-Variable Calculus, it is asking me to find the derivative of y with respect to x by implicit differentiation. The problem it gives me is ... 0 votes 0 answers 36 views ### Multivariable implicit differentiation,$U=T\frac{\partial{S}}{\partial{T}}dT$Given the equation$U=T\frac{\partial{S}}{\partial{T}}dT$where$S$is a function of V and T and V is being held constant, how do we then find$\frac{\partial U}{\partial T}$? The way I'm explaining ... • 304 -1 votes 1 answer 22 views ### Is there an order to approaching implicit differentiation questions? For example, I have the question "solve for$\frac {dy}{dx}$: $$ycos(xy)=y^2+2"$$ and I'm not sure if I should differentiate with the product rule or the chain rule first for the left side of the ... • 13 0 votes 2 answers 74 views ### Use implicit differentiation to find dy/dx for$x \ge -2$.$y\sqrt{x+2}=xy+3$; I am getting a different answer than is given $$y\sqrt{x+2}=xy+3$$ find$\frac {dy}{dx}$for$x \ge -2$. The given answer is $$-\frac{y(2(x+2)^{1/2}-1)}{2(x(x+2)^{1/2}-x-2)}$$ Here's my work so far: $$y\sqrt{x+2}=xy+3$$ $$y\frac d{dx}\sqrt{x+2}+\... • 13 2 votes 3 answers 405 views ### What happens when we take the derivative of an implicitly defined function? Consider the function y(x) implicitly decided by this equation: \sin(xy)=\cos(x+y). The graph on plane looks like this: According to my textbook, we can take the derivative of both sides with ... • 143 3 votes 4 answers 418 views ### How do I find the horizontal tangents to the curve x^3 + y^3 = 6xy? How do I find the horizontal tangents to the curve x^3 + y^3 = 6xy? James Stewart in section 3.6 of the 7th edition (on page 167) shows in a straightforward way that there's a horizontal tangent ... • 695 2 votes 1 answer 231 views ### Chain rule (multivariate) and implicit differentiation problem I have been worked out a solution for the following problem, but I am wondering if there is an easier way to solve it. I would be very grateful for any suggestions! Let z=f(x,y) a function of two ... • 567 1 vote 1 answer 44 views ### If y=f(x) such that (x+1)^3+(y+1)^3=16 then evaluate : \frac{d}{dx}\left(f(x+g(x)).g(x+f(x))\right) at x=-1+\sqrt[3]{15} Let f:\mathbb{R}\to\mathbb{R};y=f(x) be an explicit function defined by the implicit equation$$(x+1)^3+(y+1)^3=16$$Let g be the inverse of f. Evaluate :$$\frac{d}{dx}\left(f(x+g(x)).g(x+f(x))\... • 9,599 3 votes 1 answer 296 views ### Finding the area surrounded by a part of the implicit equation$\sin (y^x) = \cos (x^y)$Finding the area surrounded by the part of the implicit equation$\sin (y^x) = \cos (x^y)$such that$y\le 2n-x$where$n$is the solution to$n^n=\frac{\pi}{4}$where$n<0.5$bounded by the$x$... • 1,728 1 vote 0 answers 32 views ### Computation of smooth Cholesky factorizations Assume$X(t)$is a time dependent positive definite symmetric matrix which satisfies the matrix differential equation$\dot{X}(t) = Q(X(t)), X(0) = 0, $where$Q(X(t))\$ is a symmetric positive ...
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I have the following function $$a\text{log}\left(X\right)+b\text{log}\left(y\right)=c\left[d\text{log}\left(X\right)+e\text{log}\left(y\right)\right]+\text{log}\left(\frac{A}{B}\right)$$ I wish to ...