TABLE OF CONTENTS
Find given that
x2 + y2 = 4.
We don't have a formula for y. We could solve this equation for y, but we would find
which is two equations. We would then take two separate derivatives, and that's too much work. Instead, take derivatives of both sides of the equation, with respect to x:
Since the derivative of a sum is the sum of the derivatives, and the derivative of a constant is 0,
Pause for a minute. While we don't have a formula for y, we know that y is a function of x. That is, y is the inside function and (□)2 is the outside function.
To take the derivative of y2 we need to use the chain rule:
Put this back in the equation where we left off:
2x + d/dx(y2) = 0
2x + 2y dy/dx = 0
It's like magic. In the original equation we had a messy y2 term, but now we have dy/dx. Solving for the derivative, we find
2y dy/dx = -2x
dy/dx = -2x/2y
In this type of problem, it's fine to have y in the definition of the derivative. We can't write y in terms of x since we don't have a formula for y in the first place. The final answer is
dy/dx = -2x/2y = -x/y
Find the derivative of y with respect to x given that
4y2 + 8y = 2x2.
This equation would be horrible to solve for y, so we won't. Instead, take derivatives from here:
Again, y is a function of x, we need to use the chain rule for any derivatives involving y:
We have two occurrences of , but that's find. We can factor them out:
and then we simplify:
Assuming y is a function of x and
xy + x = y,
Take derivatives of both sides to find
Split up the derivative of the sum into a sum of derivatives to find
The derivative of x with respect to x is 1, and the derivative of y with respect to x is , we can rewrite the equation as
Since x is a function of x and y is a function of x, we need to use the product rule to find the derivative of xy.
Again, and , we find
Now we want to solve for , move all the terms with that on one side and factor:
and divide by (1-x) to find the final answer:
Make it rain.