### 6.4 Implicit Differentiation

```MATH 151 SI Name 6.4-­β6.5 6.4 Implicit Differentiation How to find dy/dx implicitly? 1. Take the derivative of the function: Example π¦ ln π₯ + 4π₯ = 13π¦ Derivative aaa All normal rules apply (power, product, !
quotient, exponential, log, chain) β’
Did you take the derivative of y? Multiply by dy/dx o
Did you take the derivative of x? β’
Leave this as is (or multiply by o
dx/dx=1) 2. Move all terms with a dy/dx in them to one side of the equation and all terms without dy/dx to the other side β’
You may need to expand or get rid of your fractions 3. Factor out dy/dx and solve for dy/dx 2π¦ ln π₯ β
+
!!
!
+ 4 = 13 β
!"
!"
ππ¦ π¦ !
ππ¦
+
+ 4 = 13 β ππ₯
π₯
ππ₯
π¦!
ππ¦
ππ¦
+ 4 = 13 β
β 2π¦ ln π₯ β π₯
ππ₯
ππ₯
by dividing by what is left !"
You took the derivative of y so multiply by dy/dx 2π¦ ln π₯ β
!"
π¦!
ππ¦
+4=
(13 β 2π¦ ln π₯) π₯
ππ₯
ππ¦
π¦!
4
=
+
ππ₯ π₯ 13 β 2π¦ ln π₯
13 β 2π¦ ln π₯
```