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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.
\[y=\sqrt { x+\sqrt { x+\sqrt { x+\sqrt { x+\cdots }}}}\]
For \( x>0 \) , define \(y\) as above. What is \(\dfrac{dy}{dx}\)?
- \(P: \dfrac{1}{2y-1}\)
- \(Q: \dfrac{x}{x+2y}\)
- \(R: \dfrac { 1 }{ \sqrt { 1+4x } } \)
- \(S: \dfrac{y}{2x + y}\)
by
Jatin Chanchlani
\[\large x^{p}y^{q}=(x+y)^{p+q}\]
If the above equation holds true for some reals \(x\), \(y\), \(p\) and \(q\), then what is true about \(\dfrac {dy}{dx} \)?
by
Jatin Chanchlani
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.
by
Sandeep Bhardwaj
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 \(m+n\)
by
Harsh Shrivastava
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 \(a^2 + b^2\).
by
Worranat Pakornrat