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!

to plot points   for x just start at the origin (middle) and go to the left however many times it say(if the number is a positive) if the number is a negative then go right however many times it says if it is 0 go nowhere

now keep you pencil there and go up however many times it says on y(if it is positive) if it is negative then go down however many times it says. if it says 0 go nowhere.

you have just plotted a point! now when you finish plotting all of them connect them with a ruler

plot each point as (-3, 7), (0,4), (3,1), (8,-4)   in the parenthesis on the left of the comma is x and the right is y.