Using Lagrange Notation Examples

If y is a function of x and

x² + y² = 16,

find y '.

We take derivatives of both sides, using the prime notation to mean "derivative with respect to x".

(x² + y²)' = (16)'

Since the derivative of a constant is 0,

(x² + y²)' = 0

Now we go to work on the left-hand side. While we don't have a formula for y, we know that y is a function of x. Therefore
any derivatives involving y need to use the chain rule.

Now we solve for y '.

Find y ' given that y is a function of x and xy + x² + y² = 0.

Take derivatives of both sides:

Remember that any derivative involving y must use the chain rule. Also, to find the derivative of xy we need to use the product rule:

Since the derivative of x with respect to x is 1, we can replace the x' with 1 to find

y + xy' + 2x + 2yy' = 0

Now we want to move all the terms with y' on one side, and all the other terms on the other side. Then we can factor out and solve for y'.