Determine the slope of the line tangent to the curve $x^2 + y^2 = 25$ at the point $(4, 3)$.

Slope $=\answer {-4/3}$

Suppose $x$ and $y$ are related by the equation $\cos (y) = xy$. Further suppose that $x$ changes by a very small amount, represented by $dx$. Write an expression for the corresponding change in $y$.

$\answer {-y/(\sin (y)+x)}$

Suppose $x$ and $y$ represent the measures of covarying quantities and that the covariation of $x$ and $y$ is constrained by the equation $x^2 + 5y^5 = 9$. At what rate does $y$ change with respect to $x$ when $x = 2$ and $y = 1$?

$\answer {-4/25}$