# Questions tagged [implicit-differentiation]

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

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### Implicit differentiation of $\ln(x^2+xy+y^2)=1$ [closed]

Find $\frac{\mathrm{d}y}{\mathrm{d}x}$ of $\ln(x^2+xy+y^2)=1$. So for this natural log should I start by using the power rule or the the product rule of $xy$? The textbook talks about taking the ...
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### Is the Primer Introduction to Derivation of Implicit Functions Using This Definition Ambiguous?

I'm exploring the concept of implicit function derivation and parameterized curve derivation, and I've encountered some points that seem vague to me. I would appreciate any insights or clarifications ...
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### Implicit Differentiation: Proving a point is perpendicular at a certain point on two interesecting tangents.

Given the following question: Consider the curves C1 and C2 defined as follows; C1 : xy = 4 , x > 0 C2 : y^2 - x^2 = 2 , x > 0 Let P(a,b) be a unique point where the curves C1 and C2 intersect. ...
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### How do I implicitly differentiate $\frac{d}{dx}(2x(y')^2y'')$

I know that I will probably need to use the 3-way chain rule: $$(fgh)'=f'gh+fg'h+fgh',$$ but since I'm differentiating in terms of $x$, I'm very confused as to how to to "append" $y'$ terms ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 $x$ ...
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### 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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### Solving logarithmic derivative in implicit form

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 ...
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### Question about implicit differentiation in polar coordinates

I was wondering what was incorrect about this procedure that seems to lead to a contradiction. In polar coordinates its true that: $x^2 + y^2 = r^2$ If I differentiate the equation implicitly with ...
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### Question on Implicit Differentiation

I'm trying to solve this problem: "Sand being emptied from a hopper at the rate of $10$ cubic feet per second forms a conical pile whose height is always twice its radius. At what rate is the ... 1 vote
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### Coupled implicit partial differentiation

I'm struggling to determine two partial derivatives of the following two implicitly-defined, coupled equations. Given: $$x = X(x, y, E) = f_x(E) + g_x(x, y)$$ $$y = Y(x, y, E) = f_y(E) + g_y(x, y)$$ ...
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### Implicit Differentiation of $x^3+y^3=3xy$

Recently I came across a problem that asked to solve for the derivative of the following equation: $$x^3+y^3=3xy$$ I started by differentiating both sides, and I got $3x^2+3y^2\frac{dy}{dx}$ on the ... 35 views

### why implicit partial differentiation and explicit partial differentiation of z with respect to x give different results?

I had a calculus exam and the question was to take the partial of z with respect ot x if: $$xe^y+yz+ze^x=1$$ I wrote z in terms of x and y: $$z= \frac{1-xe^y}{y+e^x}$$ I took the partial of the both ...
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### What is really the TRUE definition of an implicit function?

First of all I would like to say that I have already found similar questions on stack exchange but somehow my confusion regarding the definition of an implicit function still linger. The title says it ...
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### Why can't I use the chain rule for multiple variables to differentiate the logarithm of a quotient

Say I want to find the derivative of the below function with respect to x, which equals zero. \$u(x,z) = \ln(\frac{\mathrm{x}^α + \mathrm{z}^{α}}{\mathrm{z}^α}),z = h(x) = \mathrm{(a\mathrm{x}^4 + ...
I was taking the implicit derivative of the following: $$\frac x y = x - y$$ I'll cut straight to the chase and say my answer was this: $$\frac {dy}{dx} = \frac {y - y^2} {x - y^2}$$ I've come ...