Explicit Functions: y=F(x)
Implicit Functions: F(x,y)=0

All Xβs and Yβs are on one side!

Explicit functions can always be written as implicit functions

When looking for the rate of change of an implicit function, we have to to implicit differentiation
Implicit Differentiation
Example
Remember, that y=f(x), or whatever term you are differentiating in terms of.
 Where we are differentiating by dx, Y acts as a functions f(x).
 From there, we chain rule our f(x) function, by derivative of outside times derivative of inside
Example
 Here, look at the tree!
 F is a function of x, y and z
 X stands on its own (first term)
 Y does not depend on X! So it has zero rate of change (second term)
 Z depends on both X and Y, but we are differentiating in terms of X, so itβs all we care about! (Last term)
 Rearrange and solve for dz/dx!
Example to try for later!
$z_{2}y+3y_{2}xβ4x_{3}z=0$ Solve for (dx/dz)
Gradients and Directional Derivatives
Gradients
 Gradient of a function is a vector whose components are partial derivatives!
$β_{f}$ (The upside down triangle is known as the βnablaβ, which is the gradient of F)
2D
$f(x,y)=β_{f}=(βxβfβ,βyβfβ)$3D
$f(x,y,z)=β_{f}=(βxβfβ,βyβfβ,βzβfβ)$Example
w = f(x,y,z) = xy +cos(z)
$β_{w}=(βxβwβ,βyβwβ,βzβwβ)$ $=y,z,βsin(z)$Directional Derivatives
 Rate of change of F in a direction of Γ»
 Γ» can be anything chosen!
Generally,
$D_{u}f=β_{f}βu^$Example
 Gradients are perpendicular to your level curves, and point to higher levels!
 Unless the directional derivative is NEGATIVE. It then means you are pointing downwards
 Which in this case, (6/5)^{th} is.
Hereβs a slightly squished graph of that problem!
 Blue lines are the level curves lines
 Green lines are the upwards gradient lines
 Orange arrows are the Γ» vectors calculated
Some quick questions about it..

What is the direction of steepest ascent here? Parallel to the gradient, going upwards! Direction of gradient β (2x, 2y) Iβm at (1,1) β (2,2)

What is the direction of steepest decent here? Antiparallel to the gradient, going downwards! Direction of gradient β (2x, 2y) Iβm at (1,1) β (2,2)

What is the direction of no ascent or decent here? Perpendicular to gradient tangent to level curve $u^ββf=0$ Youβll have F, and youβll need to solve for Γ»
The equation of a plane that contains the point (x0,y0,z0) with normal vector βn=β¨a,b,cβ© is given by,
a(xβx_{0})+b(yβy_{0})+c(zβz_{0})=0