Question

Please help Calculus The graph of x^2=-2+y+5cosy is shown for y=11, find the derivative dy/dx

Answer 1

Taking the derivative of both sides with respect to $x$ gives

$\dfrac{\mathrm d}{\mathrm dx}[x^2]=\dfrac{\mathrm d}{\mathrm dx}[-2+y+5\cos y]$

$\implies2x=\dfrac{\mathrm dy}{\mathrm dx}-5\sin y\,\dfrac{\mathrm dy}{\mathrm dx}$

$\implies\dfrac{\mathrm dy}{\mathrm dx}=\dfrac{2x}{1-5\sin y}$

When $y=11$, we have two possible values of $x$:

$x^2=-2+11+5\cos11\implies x=\pm\sqrt{9+5\cos11}\approx\pm3.004$

so we have two possible values $\dfrac{\mathrm dy}{\mathrm dx}\approx\pm1.001$.