Question

For the graph of cos(y)=x-4y, what is the range of the slopes of its tangent lines?

Answer 1

Any tangent line to the curve has a slope given by $\dfrac{\mathrm dy}{\mathrm dx}$.

$\cos y=x-4y\implies-\sin y\,\dfrac{\mathrm dy}{\mathrm dx}=1-4\,\dfrac{\mathrm dy}{\mathrm dx}$

$\implies\dfrac{\mathrm dy}{\mathrm dx}=\dfrac1{4-\sin y}$

Since $|\sin y|\le1$, the denominator ranges from 3 to 5, so the range of slopes is $\dfrac15\le\dfrac{\mathrm dy}{\mathrm dx}\le\dfrac13$.