Find dy/dx by implicit differentiation for ysin(y) = xcos(x)

Step-by-step explanation:

So we have:

And we want to find dy/dx.

So, let's take the derivative of both sides with respect to x:

Let's do each side individually.

Left Side:

We have:

We can use the product rule:

So, our derivative is:

We must implicitly differentiate for y. This gives us:

For the sin(y), we need to use the chain rule:

Our u(x) is sin(x) and our v(x) is y. So, u'(x) is cos(x) and v'(x) is dy/dx.

So, our derivative is:

Simplify:

And we are done for the right.

Right Side:

We have:

This will be significantly easier since it's just x like normal.

Again, let's use the product rule:

Differentiate:

So, our entire equation is:

To find our derivative, we need to solve for dy/dx. So, let's factor out a dy/dx from the left. This yields:

Finally, divide everything by the expression inside the parentheses to obtain our derivative:

And we're done!