Derivative of an Implicit Function

Differentiate the given function.

Write the derivative of x²: 2x¹.

Derivative of a Polynomial

See y as a variable
and write the derivative of y²: 2y¹.
And write the derivative of y: y'.

It's like the derivative of a composite function:
First differentiate the outer function y²: 2y¹.
Then differentiate the inner function y: y'.

For this reason,
when differentiating a term with y,
multiply y'.

The right side is a constant.
So write = 0.

Then you get 2x¹ + 2y¹⋅y' = 0.

Write this equation as y' = ... .

2x¹ = 2x
+2y¹⋅y' = +2yy'

Divide both sides by 2.

Move x to the right side.

Divide both sides by y.

Then y' = -x/y.

So dy/dx = y' = -x/y.

Differentiate the given function.

Write the derivative of x³: 3x².

Write the derivative of +xy².

Write,
the derivative of x, 1 times y²
plus
x times, the derivative of y², 2y¹⋅y'.

Derivative of a Product

Don't forget to multiply y'
when writing the derivative of y²: 2y¹⋅y'.

Write the derivative of -2y³:
-2⋅3y²⋅y'.

Don't forget to write y'.

+2 is a constant.
So write 0.

The right side is 0.
So write = 0.

So 3x² + 1⋅y² + x⋅2y¹⋅y' - 2⋅3y²⋅y' + 0 = 0.

Write this equation as y' = ... .

+1⋅y² = +y²
+x⋅2y¹⋅y' = +2xyy'
-2⋅3y²⋅y' = -6y²y'

Move 3x² + y² to the right side.

And combine +2xyy' - 6y²y':
+y'(2xy - 6y²).

Common Monomial Factor

Then y'(2xy - 6y²) = -3x² - y².

Divide both sides by (2xy - 6y²).

Then y' = (-3x² - y²)/(2xy - 6y²).
This is dy/dx.

It says to find dy/dx at (1, 1).

So find [y']_{(x, y) = (1, 1)}.

Put (x, y) = (1, 1) into y' = (-3x² - y²)/(2xy - 6y²).

Then [y']_{(x, y) = (1, 1)} = (-3⋅1² - 1²)/(2⋅1⋅1 - 6⋅1²).

Then you get 1.

So dy/dx = y' = 1
at (1, 1).