Calculus

Implicit Differentiation

Implicit Differentiation: Level 3 Challenges

         

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

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

  • P:12y1P: \dfrac{1}{2y-1}
  • Q:xx+2yQ: \dfrac{x}{x+2y}
  • R:11+4xR: \dfrac { 1 }{ \sqrt { 1+4x } }
  • S:y2x+yS: \dfrac{y}{2x + y}

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

If the above equation holds true for some reals xx, yy, pp and qq, then what is true about dydx\dfrac {dy}{dx} ?

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

{x33xy2=a3yx2y3=b.\begin{cases} \begin{aligned} x^{3}-3xy^{2}&=a \\ 3yx^{2}-y^{3}&=b. \end{aligned} \end{cases}

Details and Assumptions:

  • aa and bb 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=x+12x36.y = \sqrt{x+\sqrt{12x-36}}.

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

Find the value of m+n.m+n.

As shown above is the elliptical graph of x2+xy+y2=7x^2 + xy + y^2 = 7, where P1P_{1} and P2P_{2} are the points on the graph with minimum and maximum xx coordinates respectively, and the red line ll is the linear graph passing through P1P_{1} and P2P_{2}.

If the tangent parallel to line ll touches the graph at a point P3(a,b)P_{3} (a,b) for some real numbers a,ba,b, compute a2+b2a^2 + b^2.

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