Implicit differentiation and inverse trigonometric functions

When the equation $x^{2} + y^{3} = 7$ is differentiated with respect to $x$, the term $y^{3}$ becomes $3y^{2}\,\dfrac{dy}{dx}$. Which rule produces the factor $\dfrac{dy}{dx}$?

The point $(3, 4)$ lies on the circle $x^{2} + y^{2} = 25$. Use implicit differentiation to find the slope of the tangent line there.

Let $f(x) = \arcsin\!\left(\dfrac{x}{3}\right)$. Find $f'(0)$.

The point $(1, 2)$ lies on the curve below. Find $\dfrac{dy}{dx}$ at that point.

Given $\sin(y) + x^{2} = y$, which expression is $\dfrac{dy}{dx}$?

The curve below passes through exactly one point with $y = 0$ and $x > 0$. Find $\dfrac{dy}{dx}$ at that point.

Let $g(x) = \bigl(1 + x^{2}\bigr)\arctan x$. Find $g'(1)$.

Verify that the point $(2, 4)$ lies on the curve below, then find the equation of the tangent line there.

The function $f(x) = x^{3} + 2x + 1$ is one-to-one. Find $\bigl(f^{-1}\bigr)'(4)$.

The curve $x^{2} + xy + y^{2} = 3$ has exactly two points at which its tangent line is horizontal. Find the distance between those two points.