Implicit Differentiation

A function can be explicit or implicit:

Explicit: y = f(x). In this case, f(x) is composed of one variable, x.

Examples include:

y = sin(x)
y = x² + 3
y = ln(x) + 3x + 2

Implicit: In this form, the function is expressed in terms of both y and x.

Examples include:

25 = x² + y²
4 = x³ + y⁴ + 2y + 3

y is treated as a function and must be differentiated with respect to x.

Example 1 – what is the derivative of y?

In the context of implicit differentiation, the derivative of y would be y’ or dy/dx.

Example 2 – what is the derivative of y²?

y is the inside function; (u)² is the outside function. Applying the chain rule,

g’(x) * f’(g(x)) = y’ * 2(y)² = y’*2y²

Steps:

1. differentiate both sides of the equations
2. collect all the y’ terms on one side
3. solve for y’

Example 3 – Differentiate 25 = x² + y²

Differentiate both sides.

0 = 2x^2-1 + y’ * 2(y)² = 2x + y’*2y²

Collect the y’ terms on one side and solve for y’.

0-2x = 2x-2x+ y’*2y²
-2x = y’ * 2y²

-2x /2y² = y’ * 2y²/2y²
-x/y² = y’

Example 4 – Differentiate 4 = x³ + y⁴ + 2y + 3

Differentiate both sides.

0 = 3x² + 4y³*y’ + 2y + 0

Collect the y’ terms on one side and solve for y’.

0-3x²-2y= 3x²-3x²+4y³*y’+2y-2y +0
-3x²-2y=4y³*y’

(-3x²-2y)/4y³=4y³/4y³*y’
(-3x²-2y)/4y³=y’