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Answer:
Step-by-step explanation:
So we have the equation:
And we want to find dy/dx.
So, let's take the derivative of both sides:
Let's do each side individually.
Left Side:
We have:
We can use the chain rule, where:
Let u(x) be tan(x). Then v(x) is (x-y). Remember that d/dx(tan(x)) is sec²(x). So:
Differentiate x like normally. Implicitly differentiate for y. This yields:
Distribute:
And that is our left side.
Right Side:
We have:
We can use the quotient rule, where:
f is y. g is (8+x²). So:
Differentiate:
And that is our right side.
So, our entire equation is:
To find dy/dx, we have to solve for y'. Let's multiply both sides by the denominator on the right. So:
The right side cancels. Let's distribute the left:
Now, let's move all the y'-terms to one side. Add our second term from our left equation to the right. So:
Move -2xy to the left. So:
Factor out a y' from the right:
Divide. Therefore, dy/dx is:
We can factor out a (8+x²) from the denominator. So:
And we're done!