Let $f: R \rightarrow R$ be a differentiable function and $f(1)=4$. Then the value of $\lim _{x \rightarrow 1} \int_4^{f(x)} \frac{2 t}{x-1} dt$, if $f^{\prime}(1)=2$ is

Find $\frac{dy}{dx}$ for the equation $ax + by^2 = \cos y$.

If $x > 0$ and $x^y = e^{x-y}$, then $\frac{dy}{dx}$ is equal to

If $e^y + xy = e$,the ordered pair $\left(\frac{dy}{dx}, \frac{d^2y}{dx^2}\right)$ at $x = 0$ is equal to

Let $f : (0, \infty) \rightarrow (0, \infty)$ be a differentiable function such that $f(1) = e$ and $\lim_{t \rightarrow x} \frac{t^{2} f^{2}(x) - x^{2} f^{2}(t)}{t - x} = 0$. If $f(x) = 1$,then $x$ is equal to

If ${x^2} + {y^2} = t - \frac{1}{t}$ and ${x^4} + {y^4} = {t^2} + \frac{1}{t^2}$,then ${x^3}y\frac{dy}{dx} = $