If the slope of a tangent to the curve $xy + ax + by = 0$ at the point $(1, 1)$ on it is $2$,then $a - b$ is

If $y=y(x)$ is an implicit function of $x$ such that $\log _{e}(x+y)=4 x y$,then $\frac{d^{2} y}{d x^{2}}$ at $x=0$ is equal to .... .

Find $\frac{dy}{dx}$ for the equation $\sin^{2} x + \cos^{2} y = 1$.

Let $f : R \rightarrow R$ and $g : R \rightarrow R$ be two non-constant differentiable functions. If $f^{\prime}(x) = e^{(f(x)-g(x))} g^{\prime}(x)$ for all $x \in R$,and $f(1) = g(2) = 1$,then which of the following statement$(s)$ is (are) $TRUE$?

If $f(x)$ is an invertible and twice differentiable function satisfying $f'(x) = \int_{0}^{f(x)} f^{-1}(t) dt$ for all $x \in R$ and $f'(0) = 1$,then $f'(1)$ is equal to:

If ${x^y} = {y^x},$ then $\frac{dy}{dx} = $