Using Leibniz Notation Examples

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:

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

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

What is 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 fine. We can factor them out:

and then we simplify:

All done.

Assuming y is a function of x and

xy + x = y,

what is ?

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 , so 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 , so we find

Now we want to solve for , so move all the terms with that on one side and factor:

and divide by (1 – x) to find the final answer: