If $\sin ^2 x + \cos ^2 y = 1$,then $\frac{dy}{dx} = $

If $f(x) = \frac{1}{x^2} \int_3^x (2t - 3f'(t)) dt$, then $f'(3)$ is equal to

If $x^2 y^2 = \sin^{-1} x + \cos^{-1} x$,then $\frac{dy}{dx}$ at $x = 1$ and $y = 2$ is

Let $f$ be a non-negative function in $[0,1]$ and twice differentiable in $(0,1) .$ If $\int_{0}^{x} \sqrt{1-\left(f^{\prime}(t)\right)^{2}} \,d t=\int_{0}^{x} f(t) \,d t$ for $0 \leq x \leq 1$ and $f(0)=0$,then $\lim_{x \rightarrow 0} \frac{1}{x^{2}} \int_{0}^{x} f(t) \,d t$ is:

Find $\frac{dy}{dx}$ for the function $x^{y} + y^{x} = 1$.

If $8 f(x)+6 f\left(\frac{1}{x}\right)=x+5$ and $y=x^2 f(x)$,then $\frac{d y}{d x}$ at $x=-1$ equals