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Answer:
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!