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 = x^{2} + 3

y = ln(x) + 3x + 2

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

Examples include:

25 = x^{2} + y^{2}

4 = x^{3} + y^{4} + 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 y ^{2}?**

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’*2y^{2}

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 = x ^{2} + y^{2}**

Differentiate both sides.

0 = 2x^{2-1} + y’ * 2(y)^{2} = 2x + y’*2y^{2}

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

0-2x = 2x-2x+ y’*2y^{2}

-2x = y’ * 2y^{2}

-2x /2y^{2} = y’ * 2y^{2}/2y^{2}

-x/y^{2} = y’

**Example 4 – Differentiate 4 = x ^{3} + y^{4} + 2y + 3**

Differentiate both sides.

0 = 3x^{2} + 4y^{3}*y’ + 2y + 0

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

0-3x^{2}-2y= 3x^{2}-3x^{2}+4y^{3}*y’+2y-2y +0

-3x^{2}-2y=4y^{3}*y’

(-3x^{2}-2y)/4y^{3}=4y^{3}/4y^{3}*y’

(-3x^{2}-2y)/4y^{3}=y’

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