Question

The graph of cosy=x-4y, what is the range of slopes of its tangent lines?

Answer 1

\[
\cos y=x-4y \\
\text{Differentiate implicitly:} \\
-\sin(y) y' = 1 - 4y' \\
4y' - \sin(y)y' = 1 \\
y'(4 - \sin(y)) = 1 \\
y' = \frac{1}{4 - \sin(y)} \\
\text{Range of }\sin(y) \text{ is: }[-1, 1] \\
\text{Therefore, range of }y' \text{ is: }\frac{1}{4 + 1} \text{ to } \frac{1}{4 - 1} =
\large \left [\frac 15, \frac 13 \right ]. \\
\]