For questions on finding and evaluating derivatives when a function is defined implicitly.
Questions tagged [implicit-differentiation]
1329 questions
1
vote
1 answer
How do I work out this rate?
The problem says — Water is running into a conical reservoir, $10$ cm deep and $5$ cm in radius at the rate of $1.5$ c.c. per minute.
1) At what rate is the water level rising when the water is $4$ cm deep?
2) At what rate is the area of the water…
user733666
- 578
1
vote
2 answers
Trouble finding solution via implicit differentiation
I'm currently using the book University Calculus: Alternate Edition (Hass et al., 2008) to study calculus. I was studying implicit differentiation and ran into an exercise problem that I was having trouble solving. The question is:
Use implicit…
Sean
- 1,487
1
vote
0 answers
Fly traveling a through a point, along curve of intersection
The temperature in 3-space is given by:
$$
T(x,y,z)= \frac12(2x^2+5y^2+4z^2)
$$
At time $$t = 0,$$ a fly passes through the point $$(\sqrt{15},\sqrt{10},5),$$ flying along the curve of intersection of the surfaces
$$
z = x^2-y^2
$$
and
$$
z^2 =…
Martin Pham
- 41
1
vote
1 answer
Implicit differentiation of an equation of a hyperbola
Prove that an equation of the tangent line to the graph of the hyperbola :
$(x^2/a^2) - (y^2/b^2) = 1$
at the point ($x_0$, $y_0$) is
$x x_0/a^2 - y y_0/b^2 = 1$ (1)
I implicitly differentiated the equation and then found the gradient by…
user639649
- 395
1
vote
2 answers
Is this the correct way to do Implicit Differentiation?
Problem:
Use implicit differentiation to compute $\frac{\partial z}{\partial x}$ and $\frac{\partial z}{\partial y}$ of the function $ x^3 + y^3 +z^3 - 3xyz = 0 $.
What I Got:
$3x^2 + 0 + 3z^2 \cdot \frac{\partial z}{\partial x} - 3y \cdot…
user430574
1
vote
2 answers
Related Rates: Ladder Problem
A $13$ foot ladder is leaning against a wall. If the top slips down the wall at a rate of $4 ft/s$, how fast will the foot be moving away from the wall when the top is $10$ feet above the ground?
I got $\frac{-8\sqrt{69}}{20}$ ft/s but this doesn't…
Peter Ferri
- 49
1
vote
1 answer
finding the number of critical points of an implicitly defined equation
How do you find the number of critical points of an implicit equation such as $xy(x-6y)=9a^3$ ?
I have managed to differentiate and get $$\frac{dy}{dx} = 6y^2 - 2xy.$$ I don't know if I'm on the right route.
Jaimie1000
- 13
1
vote
0 answers
Find the value of $\theta$ that maximizes $t_c$.
There's a point $A(4t\sin\theta,4t\cos\theta)$ (where $\theta$ is a constant),at moment of $T=t$,constantly moving.
Also Let there be points $B_1,B_2,B_3$whose coordinates are$B_1(10,0),B_2(-5,5\sqrt3),B_3(-5,-5\sqrt3)$ when $T=0$.
These points…
1
vote
0 answers
Implicit function Taylor series error bound
Given the implicit curve
$$x^4y^5+x^2y^3+y+x-1=0$$
I have found the Taylor polynomial around $x=0$ of third order to be
$$p(x)=1-x-x^2+3x^3.$$
My objective now is to find an error bound for the approximation for $y(0.1)$ given by $p(0.1)=0.893$. Of…
Ray Bern
- 627
1
vote
1 answer
finding $du/dx$ if $u=u(x)$ is defined with system of equations
Assume function $u=u(x)$ is defined with that system of equations:
$$
\begin{cases}
u=f(x,y,z)\\
g(x,y,z)=0\\
h(x,y,z)=0
\end{cases}
$$
How can i find $du/dx$?
Please help, i don't know how to start solving this..
Lu Vue
- 89
1
vote
2 answers
Find the derivative $y=({1\over x})^x$
I got $y'= x^2 + \ln({1\over x})\times ({1\over x})^x$
Am I correct?
So $y'=-x^{-x}(\ln{x} + 1)$
dsta
- 335
1
vote
2 answers
Find $y'$ if $\sin(x+y) = \cos(xy)$
Find $y'$ if $\sin(x+y) = \cos(xy)$. I got ${-\cos(x+y)\over \sin(xy) * (x+y)}$. Please use Implicit Differentiation. Did I get the answer right?
dsta
- 335
1
vote
3 answers
Find an equation for a tangent line to the curve $x^2- y^2=5$ that passes through the point $(1, 1)$.
Find an equation for a tangent line to the curve $$ x^2 - y^2 = 5$$ that passes through the point $(1, 1)$.
I realize that I have to use implicit differentiation $$2x - 2y \frac {dy}{dx} = 0$$
$$\frac {dy}{dx} = \frac xy$$
However I do not know how…
Isaiah Banta
- 57
1
vote
6 answers
Implicitly differentiate $xy=1$
$xy = 1 \implies y = 1/x$, which means that $\mathrm dy/\mathrm dx$ should equal $-x^{-2}$ through the power rule.
How would you get this through implicit differentiation using the equation $xy = 1$?
Haim
- 757
1
vote
2 answers
Equation of a line tangent to ellipse
Given equation $x^2+9y^2=81$ and the point $(27,3)$, find the equation of 2 lines that pass through the point $(27,3)$, and is tangent to the ellipse
so by using implicit differentiation I got $y'=\frac{-x}{9y}$, which is the slope of the line. but…
