Makeup #7

Implicit Derivatives

The only difference between implicit derivatives and regular derivatives is that implicit derivatives include dy or y', the actual derivative of y.

y=x+2 y'=1

In an implicit derivative, you are always asked to solve for y'.

Example:

x^2+2y=0

1. Take derivative of both sides first.

2x+2y'=0

2. Then solve for y'.

y'=(-2x)/2

Some examples include:

4x+13y^2=4 y'=(-4/26y)

cos(x)=y y'=-sin(x)

y^3+y^2-5y-x^2=4 y'=2x/((3y+5)(y-1))



Some Derivative Formulas just for refreshing:

d/dx c=0 (c is a number)
d/dx cu=cu' (c is a number)
d/dx cx=c (c is a number)
d/dx u+v=u'+v' (also works the same for subtraction)
d/dx uv=uv'+vu'
d/dx u/v=(vu'-uv')/v^2
d/dx sinx=cosx(x')
d/dx cosx=-sinx(x')
d/dx tanx=sec^2x(x')
d/dx secx=secxtanx(x')
d/dx cscx=-cscxtanx(x')
d/dx cotx=-csc^2x(x')
d/dx lnu= 1/u(u')
d/dx e^u=e^u(u')