Implicit Differentiation — Question 4

Question 4 - Solution

(b) Tangent slope

At $(0,0)$, the original equation reads $\sin0=0+0$, so the point belongs to the curve.

(a) Implicit derivative

Implicit differentiation gives

$$\cos(xy)(y+xy')=1+y'.$$

Collecting derivative terms,

$$(x\cos(xy)-1)y'=1-y\cos(xy).$$

Thus, wherever the denominator is nonzero,

$$\boxed{y'=\frac{1-y\cos(xy)}{x\cos(xy)-1}}.$$

At the specified point the denominator is $-1$, and the slope is $\boxed{-1}$.