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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.

Implicit Differentiation: Level 3 Challenges

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

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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