If $f(x) = (\cos x)(\cos 2x) \ldots (\cos nx)$, then $f^{\prime}(x) + \sum_{r=1}^n (r \tan rx) f(x)$ is equal to

Consider the non-constant differentiable function $f$ of one variable which obeys the relation $\frac{f(x)}{f(y)}=f(x-y)$. If $f^{\prime}(0)=p$ and $f^{\prime}(5)=q$, then $f^{\prime}(-5)$ is

Let $f : R \to R$ be a twice differentiable function satisfying $f(0) = f(1) = 0$ and $f'(x) = f^2(x)$ for all $x \in R$. Then $\lim_{x \to 2} (f(x) + xf'(x) + x^2f''(x))$ is equal to:

If $3^x y^x = x^{3y}$, then the value of $\frac{dy}{dx}$ at $x = 1$ is:

For $x>1$,if $(2 x)^{2 y}=4 e^{2 x-2 y}$,then $(1+\log 2 x)^2 \frac{d y}{d x}$ is equal to

$x\sqrt{1 + y} + y\sqrt{1 + x} = 0$,then $\frac{dy}{dx} = $