List of top Mathematics Questions on Second Order Derivative

If \(y^{1/n}+y^{-1/n}=2x\), then \((x^{2}-1)\frac{d^{2}y}{dx^{2}}+x\frac{dy}{dx}=\ldots\)

Let \(y^{2}=a\cos ^{2}x+b\sin ^{2}x\) be such that \(a+b=2024\) and \(a>0,b>0\) follows \(y^{4}+y^{3}y^{''}=k\). If \(\lambda\) is the maximum possible value of \(k\), then \(\lambda =\)

If \(y=\cos ^{-1}x\), then \(\frac{d^{2}y}{dx^{2}}=\)

If \(y=x-e^{x}\), then \(\frac{d^{2}x}{dy^{2}}=\) ...

If \(f(x)=\begin{vmatrix} x^{2} & \sin x & e^{x} \\ 2 & 1 & -1 \\ p & p^{2} & 1 \end{vmatrix}\) then \(f^{''}(0)=\)

If \(x=at^{2}\) and \(y=2at,t\neq 0\) then ...

If \[ \sqrt{x}+\sqrt{y}=\sqrt{a}, \] then \[ \left(\frac{d^2y}{dx^2}\right)_{x=a} = \]

If \(x = asin^3t\) and \(y = acos^3t\), then the value of \(\frac{d^2y}{dx^2}\) at \(t = \frac{π}{3}\) is equal to

If \(x = 4t^3+3,y = 3t^4+4\) and \(\frac{\frac{d^2x}{dy^2}}{(\frac{dx}{dy})^n}\) is constant then the value of \(n\) is

If \(x^m+y^m = k,(m\neq 1)\) and \(y^{''} = \frac{ax^b}{y^c}\) such that \(a+b+c = 0\), then the value of k is...

Let \(x = at^2-1\), where \(a > 0\) and \(y = t^3+1\). If at \(t = 1\), \(\frac{d^2y}{dx^2} = \frac{3}{16}\), then the value of \(a\) is...

If \(f(x) = cot^{-1}(\frac{3x-x^3}{1-3x^2})\) and \(g(x) = cos^{-1}(\frac{1-x^2}{1+x^2})\), then \(\underset{x\rightarrow a}{lim}\frac{f(x)-f(a)}{g(x)-g(a)}\), \((0 < a < \frac{1}{2})\) is

If \(y^{\frac{1}{m}}+y^{\frac{-1}{m}} = 2x\) , then \((x^2-1)y_1^2 =\)

If \(e^y+xy = e\), then the ordered pair \((\frac{\text{d}y}{\text{d}x},\frac{\text{d}^2y}{\text{d}x^2})\) at \(x = 0\) is equal to

If \(y = x+e^x\), then \(\frac{d^2x}{dy^2}\) is equal to

If \(x^3+y^3 = 6\) and \(\frac{d^2y}{dx^2}\cdot \frac{d^2x}{dy^2} = \frac{m}{(xy)^n}\), where \(m,n\in R\) then \(\frac{m}{n} =\)

If \(y = \sqrt{x^2+8x+3}\), then the value of \(\frac{d^2y}{dx^2}\) at \(x = 3\) is

If \( y = \cos^{-1} x \), find \( \frac{d^2y}{dx^2} \) in terms of \( y \).

If $y = 4\sqrt{x}$, then $\frac{d^2y}{dx^2} =$}

Let $y = 4e^{-x} - 2e^{-2x} - e^{-3x}, x \in \mathbb{R}$. If $\frac{d^2y}{dx^2} = e^{\alpha x}(4e^{2x} - 8e^x - 9)$ for all $x$, then the value of the constant $\alpha$ is

For \(n\in\mathbb{N}\), if \[ y=ax^{n+1}+bx^{-n}, \] then \[ x^2\frac{d^2y}{dx^2} \] is equal to:

For \(n\in\mathbb{N}\), if \[ y=ax^{n+1}+bx^{-n}, \] then \[ x^2\frac{d^2y}{dx^2} \] is equal to:

If \(y(x)=x^x,\ x>0\), then \[ y''(2)-2y'(2)= \]

The second derivative of \( \sin 3x \cos 5x \) is:

If $y=e^{mx}\sin(nx)$, then the value of \[ \frac{d^2y}{dx^2}-2m\frac{dy}{dx}+(m^2+n^2)y \] is: