# Implicit differentiation $\sin(xy)$

When I check my answer using the implicit differentiation tool on wolframalpha.com, I get a result I can't agree with. So I'd like to hear your opinion :)

Asked: use implicit differentiation to differentiate $$\sin(xy)$$.

My take on the matter:

Using the chain rule: $$\frac{d}{dx} \sin(xy)$$ = $$\cos(xy) \frac{d}{dx} xy$$, then using the product rule on latter factor we get: $$\frac{d}{dx} xy = y + x\frac{dy}{dx}$$

Hence: $$\frac{d}{dx}\sin(xy) = y \cos(xy) + x \cos(xy) \frac{dy}{dx}$$.

Is this correct? Wolframalpha tells me it's simply: $$y \cos(xy)$$, but perhaps I'm using the tool wrong...

Depending on how you entered it into WolframAlpha, it is most likely partial differentiating, meaning it considers $$x$$ and $$y$$ to be independent variables (specifically, $$\frac{\partial y}{\partial x} = 0$$). This is why WA's result is different from yours. If you want WA to interpret $$y$$ as a function of $$x$$, you have to write y(x), not just y when you enter $$y$$ into the expression.
I think you have the function $$f(x)=\sin(xy(x))$$, where $$y$$ is a differentiable function of $$x$$.
If you have the function $$g(x,y)=\sin(xy)$$ of two variables, then Wolfram computed the partial derivative $$g_x.$$