For questions on finding and evaluating derivatives when a function is defined implicitly.
Questions tagged [implicit-differentiation]
1329 questions
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Comparative Statics, implicit functions and second order derivatives.
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 $\frac{\partial x^{*}(w,z)}{\partial w}$, $\frac{\partial…
Karl Smith
- 21
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Implicit Differentiation wrt t.
Suppose $x=x(t)$ and we want to differentiate wrt t. What would be the derivative of $(1-t)x^2=t^3$? Here is my try: $\frac{d}{dt} ((1-t)x^2=t^3) = (-x^2 + (1-t)\frac{d}{dt}x^2 = 3t^2) = (-x^2 + 2(1-t)x' = 3t^2)$. But on my answer sheet we have…
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Intuitive explanation of the sign of the derivative of an implicit function.
I have three functions $f(y), g(y)$ and $h(x)$. I know that all three are positive valued and the first is increasing and the last two are decreasing. I also know that
$$ g(y)=\frac{f(y)}{h(x)}.$$
The implicit funtion $y=F(x)$ is decreasing and I…
Patricio
- 1,604
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differentiation of explicit function vs implicit function
I have the following implicit function: $xy=5$.
I want to get $y'$ or $\frac{dy}{dx}$.
In this case, implicit differentiation gives me $\frac{dy}{dx}=\frac{-y}{x}$.
I am curious why explicit differentiation cannot generate the same answer.
That is,…
jck21
- 141
- 4
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Implicit Function Theorem
Show that the equations
$$2e^x + yu - 4v + 3 = 0$$
$$y \cos x - 6x + 2u - w = 0$$
can be solved for functions
$$x = f_1(u,v,w)$$
$$y = f_2(u,v,w)$$
in a small ball with centre $(3,2,7)$ such that $f_1(3,2,7)=0$, $f_2(3,2,7)=1$.
Risa
- 303
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Implicit Function Theorem
a) Show that the equations
\begin{align*}
x^3-y^2+2u-v&=1,\\
x^2+y^3+u-v&=2
\end{align*}
can be solved simultaneously for functions $u=g(x,y),\ v=h(x,y)$ in a neighbourhood of $(1,1)$ such that $g(1,1) = h(1,1) = 1$.
b) Calculate $\frac{dg}{dx}$ and…
Risa
- 303
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1 answer
What is an implicit function?
According to the definition of implicit function we cannot determine the value of one variable explicitly from the function. I have gone through many websites, few books and few youtube videos but could not understand implicit function…
MSKB
- 305
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1 answer
How to find a tangent when you have an implicit equation and the e function
Consider this:
Find the tangent to the function $f$ at point $(2,f(2))$ where $f(2)=0$ and the function $f:(1,\infty)\to\mathbb{R}$ using the implicit equation:
$$xe^{xf(x)} = (x-2)^3 + e^{f(x)}+1.$$
Now I've found tried to find the $g'(x)$ and…
codeisfun
- 119
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2 answers
How do I differentiate a differential?
Consider:
$$ dU = T dS - PdV$$
Now, how would I differentiate both sides with temperature...?
Or more simply:
$$ dy = f'(x) dx$$
Can I differentiate this expression above? (I'm not talking about moving dx to the denominator on left)
More…
tryst with freedom
- 11,538
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votes
1 answer
Implicit differentiation of $\log$ and $\sin$ function
I am struggling with the following problem for implicit differentiation.
I am tasked to differentiate implicitly the following function, and evaluate $y''(0)$, where $y=y(x)$.
$$\ln(y+1)+\sin(xy)=\ln(5).$$
I have differentiated this once to…
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To prove a function is differentiable under certain condition
A function $f:(0, \infty) \to \Bbb{R} $ satisfies the condition $f(xy) =f(x) +f(y) $ for all $x>0,y>0$. If $f$ is differentiable at $1$, prove that $f$ is differentiable at every $c \in(0,\infty)$ and $f'(c) = \frac{1}{c}f'(1)$.
My…
Math-Learner
- 732
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1 answer
Find partial derivative of $\displaystyle u= {e}^{ -2x }\cos 4y$ and $\displaystyle v = { e }^{ -2x }\sin 4y$
Let $\displaystyle u= {e}^{ -2x }\cos 4y$ and $\displaystyle v = { e }^{ -2x }\sin 4y$
Use implicit differentiation to evaluate $\displaystyle \frac { \partial x }{ \partial u }$ at $(x,y)= (1,2).$
I'm not really sure how to do this one.
Dexter
- 189
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4 answers
How to find the implicit differentiation of a fraction?
I need to determine the whether point P is a local max/min or stationary point. So I need to take the second derivative.
Question:
$5x^2+6xy+5y^2 = 8$
I figured out that the first derivative is:
$\frac{dy}{dx} = \frac{-10x-6y}{6x+10y}$
Therefore…
CountDOOKU
- 1,065
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votes
1 answer
Find the Co-ordinates of the stationary points on the curve $f(x)=2x^3-4x^2+2$
Find the Co-ordinates of the stationary points on the curve $$f(x)=2x^3-4x^2+2$$
What I did was to differentiate $f(x)$ then factorise to find to possible $x$ values then put those two values into $f(x)$.
Although I came out with $$f(0) = 2,$$…
jackdh
- 161
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4 answers
gradient at $(0,0)$ of $3y^{2 }=2x^{3\ }+x^{2}$
Probably missing something simple, but how do I find the gradient of: $3y^{2 }=2x^{3\ }+x^{2}$ at (0,0)?
I get derivative:
$6y\frac{dy}{dy} =6x^2 +2x$, and when I stick in (0,0) into this, it's undefined, but from the graph below, it looks like it…
Noobcoder
- 227