Calculus
# Implicit Differentiation

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

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

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

**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 \(a^2 + b^2\).

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