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.

Differentiation 1
0 = 2x2-1 + 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.

Differentiation 2
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’

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