Mathematics Stack Exchange
Mathematics Stack Exchange

Questions tagged [implicit-differentiation]

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

1329 questions
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Implicit differentiation in terms of x, y, r and f

I just want to check am I practicing implicit differentiation correctly. $$r = x+f(y+ rx + x^3)$$ where f is a differentiable function. I am trying to use implicit differentiation to find $\frac{\partial r}{\partial x}$ in terms of $y,\ x,\ r$ and…
number8
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Derivative of $\dfrac{dy^2}{dx}$

I just finish learning the chain rule and am now learning Implicit Differentiation and I am wondering: why is it not possible to take the derivative of $\dfrac{dy^2}{dx}$? Why do we need to apply the chain rule and find…
user5826
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Quotient rule and implicit differentiation

Find $\frac{dy}{dx}$ for $x^2=\frac{x-y}{x+y}$. I have solved this in two ways. First, I multiplicated the whole equation by $x+y$ and then I calculated the implicit derivative. I got the following solution: $\frac{1-3x^2-2xy}{x^2+1}$ So far so…
Pawel
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Trig Differentaition

Differentiate $y=27 \sec^3(x)$ with respect to $x$. I tried splitting the $\sec^3(x)$ into $\sec^2(x)\cdot \sec(x)$ and using the product rule but that didn't work.
madeye moody
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Implicit function question

Explain why $\sqrt{x^2-y^2}+\arcsin(x/y)=0$ does not define $y$ as an implicit function of $x$. Quite confused by this, mainly because I do not fully understand really what it means for an equation to define $y$ as an implicit function of $x$ even…
Carlos Bacca
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implicit differentiation yielding different expressions

I was studying Thomas's Calculus book and attempted a question using implicit differentiation. $$x^3=\frac{2x-y}{x+3y}$$ I differentiated both sides directly, using the quotient rule on the RHS to obtain. $$ \frac{dy}{dx}=\frac{7y-3x^2(x+3y)^2}{7x}…
matt
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Is this the best route to go in this implicit derivation problem?

I have this equation: $$e^{\frac{x}{y}} = x - y$$ I seem to be going down the wrong path. Is this right so far? $$\frac{dy}{dx} = e^{\frac{x}{y}} \cdot \frac{dy}{dx} (x \cdot y^{-1}) = 1 - y'$$ $$e^{\frac{x}{y}} ( x \cdot -y^{-2} \cdot y' + y^{-1})…
Jwan622
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Is this the right application of implicit derivation?

$$x^2 - 4xy + y^2 = 4$$ $$2x - 4xy' -4y + 2yy' = 0$$ $$-4xy' + 2yy' = -2x + 4y$$ $$y'(-4 + 2y) = -2x + 4y$$ $$y' = \frac{-2x + 4y}{-4x+2y}$$ But the answer on wolfram is: $\frac{x - 2y}{2x - y}$ Sorry for the newb question. Is this right? And also,…
Jwan622
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What should be a simple implicit derivative

I'm currently working my way through an economics paper which has a derivative that at the moment I'm failing to see. I was hoping that it would be a simple case of the implicit function theorem however, using that approach I don't find the correct…
mark
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How to find radius of circle given its center and the equation to an ellipse

Question: A circle is drawn with its center at (8,0) and with radius r such that the circle cuts the ellipse $x^2 + 4y^2 = 16$ at right angles. Find the radius of the circle. I understand that the tangent of a circle at the point they intersect is…
Jo Villar
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If $y=x^{(e^x)}$, then what is an expression for dy/dx?

If $y=x^{(e^x)}$, then which of the following is an expression for $\frac{dy}{dx}$? Possible answers could be: A: $e^x \ln(x)+ e^x/x$ B: $e^x x^{e^x}(\ln(x)+1/x)$ C: $x^{e^x}(e^x \ln(x) - 1/x)$ For my answer I got C but I'm not sure if that's right.…
John Smith
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Implicit Differentiation Confusion

My question is simple. We are given the following function: $\frac{x^2-y^2}{x^2+y^2}=\frac{1}{2} $. We are asked to find the derivative implicitly. If we use the chain rule on the left-hand side, we can solve for $\frac{dy}{dx} = \frac{y}{x}$. But…
dacastr
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Derivative of $(log (x))^{x}$

How can we calculate the value of $ \frac{dy}{dx} (log (x))^{x}$ I tried doing it the following way : Let $ y= (\log (x))^{x} $ $ \log y = x \log \log (x)$ Then differentiating both sides with respect to $x$ but its not working.
Mojo Jojo
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find $\frac{dy}{dx}$ in terms of $x$ and $y$ of $x^2y^2=\frac{(y+1)}{(x+1)}$ - basic question

Find $\frac{dy}{dx}$ in terms of $x$ and $y$ of $x^2y^2=\frac{(y+1)}{(x+1)}$ Ok so using the product rule on the LHS and the quotient rule on the RHS I differentiated both sides of the equation and…
Rogerc1979
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Difficulty Understanding Implicit Differentiation

I am struggling with an Implicit Differentiation question which is as follows: $z = (7x^4)*\ln(x)4$ where $z$ and $x$ are functions of $t$. $\frac{dx}{dt} = 4$ when $x = e$. Calculate $\frac{dz}{dt}$. What I have tried so far is finding…
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