$$\text { If } \sin (x+y)+\cos (x+y)=\sin \left[\cos ^{-1}\left(\frac{1}{3}\right)\right] \text {, then } \frac{dy}{dx}=$$

For every value of $$x$$, the function $$f(x)=\frac{1}{a^x}, a>$$ 0 is,

If $$f(x)=\log (\sin x), x \in\left[\frac{\pi}{6}, \frac{5 \pi}{6}\right]$$, then value of '$$c$$' by applying LMVT is

The equation of tangent at $$P(-4,-4)$$ on the curve $$x^2=-4 y$$ is