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Implicit Differentiation

Not all equations can be written like y = f(x), so taking the derivative can be tricky. Save the mess and do it directly with implicit differentiation.

Level 3

         

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

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

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

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

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

\[\begin{cases} x^{3}-3xy^{2}=a \\ 3yx^{2}-y^{3}=b \end{cases}\]

Details and Assumptions

  • \(a\) and \(b\) are real numbers.

  • Angle of intersection of curves is the angle between the tangents to the curves at the point of their intersection.

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

The value of \(\dfrac{dy}{dx}\) at \(x = 4\) can be expressed as \(\dfrac{m}{n}\) , where \(m , n \) are co-prime 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 \(a^2 + b^2\).

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