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Differentiation Implicit Differentiation
Level 8
12
IMPLICIT DIFFERENTIATION The equation of an explicit function has its dependent variable on one side of the equal sign and the independent variable(s) on the other side:
𝑦 = 𝑥$ + 2𝑥 − 1
In the equation of an implicit function dependent variable is not alone:
2𝑥$𝑦 −𝑥𝑦= 2𝑥
Recall that the derivative of the 𝑦 is 𝑦′ or 𝑑𝑦/𝑑𝑥. Particularly in the last notation, we refer to the derivative as “the derivative of 𝑦 with respect to 𝑥.” When taking the derivative of an implicit function, whenever you take the derivative of 𝑦, you must multiply by 𝑑𝑦/𝑑𝑥 or 𝑦′ (by Chain Rule). So, the derivative of 𝑥$ with respect to 𝑥 is:
𝑥$ . = 2𝑥 ∙𝑑𝑥𝑑𝑥
= 2𝑥
But the derivative to 𝑦$ with respect to 𝑥 is:
(𝑦$)′ = 2𝑦 ∙𝑑𝑦𝑑𝑥
After taking the derivative of both sides, isolate for 𝑑𝑦/𝑑𝑥 to get the derivative. This process is called implicit differentiation. EXAMPLE: Find the derivative of the circle: 𝑥$ + 𝑦$ = 25. SOLUTION: We’ll take the derivatives of both sides and isolate for 𝑑𝑦/𝑑𝑥:
2𝑥 + 2𝑦 ∙𝑑𝑦𝑑𝑥
= 0
2𝑦 ∙𝑑𝑦𝑑𝑥
= −2𝑥
𝑑𝑦/𝑑𝑥 = −𝑥/𝑦
EXAMPLE: Find the derivative of 𝑥𝑦 + 𝑦 = 2𝑥 − 1. SOLUTION: Notice that the first term is a product of two functions: 𝑥 and 𝑦. We’ll need to use product rule to differentiate:
1 𝑦 + 𝑥 ∙ 𝑦′ + 𝑦′ = 2 + 0
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Differentiation Implicit Differentiation
Level 8
13
Move 𝑦 to the right side and factor out the 𝑦′:
𝑦′ 𝑥 + 1 = 2 − 𝑦
Now isolate for 𝑦′:
𝑦′ =− 𝑦 − 2𝑥 + 1
EXAMPLE: Find the derivative of 𝑥𝑦 = 4 both implicity and explicitly. SOLUTION: We can solve for 𝑦 to obtain an explicit equation:
𝑦 =4𝑥= 4𝑥56
Then take the derivative:
𝑦. = −4𝑥5$ = −4𝑥$
Implicitly, we could use product rule:
1 𝑦 + 𝑦. 𝑥 = 0𝑥𝑦. = −𝑦𝑦. = −𝑦/𝑥
This may look different but recall that 𝑦 = 4/𝑥. We can subsitute that explicit equation into the derivative and see that it is the same expression:
𝑦. =−4𝑥𝑥
= −4𝑥$
Manipulating Functions You can manipulate functions before taking the derivative in order to avoid quotient rule and get rid of square roots. For example:
𝑦 − 1𝑥
=𝑥𝑦3
Square both sides to obtain:
𝑦$ − 2𝑦 + 1𝑥$
=𝑥𝑦9
Then cross multiply:
9𝑦$ − 18𝑦 + 9 = 𝑥:𝑦
This is an easier function to differentiate.
