Implicit Differentiation: Level 3 Challenges Practice Problems Online

$y=\sqrt { x+\sqrt { x+\sqrt { x+\sqrt { x+\cdots }}}}$

For $x>0$ , define $y$ as above. What is $\dfrac{dy}{dx}$ ?

$P: \dfrac{1}{2y-1}$
$Q: \dfrac{x}{x+2y}$
$R: \dfrac { 1 }{ \sqrt { 1+4x } }$
$S: \dfrac{y}{2x + y}$

P

,

Q

and

R

Q

,

R

and

S

Only

P

Only

Q

P

,

R

and

S

P

,

Q

,

R

and

S

$\large x^{p}y^{q}=(x+y)^{p+q}$

If the above equation holds true for some reals $x$ , $y$ , $p$ and $q$ , then what is true about $\dfrac {dy}{dx}$ ?

Find the angle (in degrees) of intersection of the curves

$\begin{cases} \begin{aligned} x^{3}-3xy^{2}&=a \\ 3yx^{2}-y^{3}&=b. \end{aligned} \end{cases}$

Details and Assumptions:

$a$ and $b$ are real numbers.
The angle of intersection of curves is the angle between the tangents to the curves at the point of intersection.

Let $y = \sqrt{x+\sqrt{12x-36}}.$

The value of $\frac{dy}{dx}$ at $x = 4$ can be expressed as $\frac{m}{n},$ where $m$ and $n$ are coprime positive integers.

Find the value of $m+n.$

As shown above is the elliptical graph of $x^2 + xy + y^2 = 7$ , where $P_{1}$ and $P_{2}$ are the points on the graph with minimum and maximum $x$ coordinates respectively, and the red line $l$ is the linear graph passing through $P_{1}$ and $P_{2}$ .

If the tangent parallel to line $l$ touches the graph at a point $P_{3} (a,b)$ for some real numbers $a,b$ , compute $a^2 + b^2$ .

Concept Quizzes

Challenge Quizzes

Implicit Differentiation: Level 3 Challenges

Excel in math and science

Master concepts by solving fun, challenging problems.

It's hard to learn from lectures and videos

Learn more effectively through short, interactive explorations.

Used and loved by over 7 million people

Learn from a vibrant community of students and enthusiasts, including olympiad champions, researchers, and professionals.