Composite, Implicit, and Inverse Functions

Which equation should be differentiated implicitly?

Given that curve defined by formula $e^{xy} + \ln(xy) = 7$ exists where $y$ is function $f$ from variable $x$ towards some real numbers closeby, how much does $f'$ from variable $x$ change when happening at coordinate $(1, \ln(2))$?

Given the relation between x and y expressed by $\sin(xy) = x - y$, what must be determined first in order to find $\frac{dy}{dx}$ at any point?

If $xy + y^2 = 1$ and we need to find $\frac{dy}{dx}$ when $x=2$ and $y=-1$, which step correctly uses implicit differentiation?

Which step of implicit differentiation involves factoring out y'?

Given the equation $x^2y - \sin(y) = x$, which method is most efficient for finding $\frac{dy}{dx}$ at the point where $x = 1$?

What is $\frac{dy}{dx}$ for a point on an ellipse described by the equation $4x^2+9xy+16y^2=-36$, where $a$, $b$, and $c$ are non-zero constants defining the ellipse shape?

Given that both functions defined implicitly by the equations $e^{xy} = x + y$ and $\ln(x+y) = xy$, intersect at right angles at a certain point P in the first quadrant, what is the product of their slopes at P?

What is the slope of the tangent line to the curve given by the equation $e^{xy}=x+y$ at the point where $x=0$?

If a curve is defined implicitly by the expression $\ln(xy) + e^{xy} = 7$, what's the second derivative $\left( \frac{d^2 y}{dx^2} \right)$ at a point where $x = y = e^{-1}$?