Differentiation

Question 1

Definition of a Partial derivative

Answer 1

lim f(a+h, b) - f(a,b) = fx
h→0_{……………..}h_{………………………………….}

Question 2

Gradient of a Function

Answer 2

∇ f (a) = (fx, fx₂, … , fx_n)

Question 3

Tangent Plane

Answer 3

Plane tangent to surface S at point P and including F_x and F_y

z - f(a,b) = f_x(a,b)(x-a)+f_y(a,b)(y-b)

Question 4

Differentiability

A

Answer 4

A function is differentiable at (a,b) if all gradients (f_x, f_y) exist

Question 4

Normal Vector

Answer 4

<x, y, z> = <a, b, f(a,b)> + t<f_x(a,b), f_y(a,b), -1>

Question 5

Differential of f

at (a,b)

Answer 5

df(a,b) = f_x(a,b)(x-a)dx + f_y(a,b)(y-b)dy

Question 6

Linear Approximation

Answer 6

L(x,y) = f(a,b) + f_x(a,b)(x-a) + f_y(a,b)(y-b)

Question 7

Implicit Differentiation

What does a questoin look like

Answer 7

x²+xy+xz+y²+yz+z²=5

find z_x, z_y

remember product rule: z_x in xz.
xz dx = xz’+z

Question 8

Chain Rule

Answer 8

df = ∂f * dx + ∂f * dy
dy = ∂x * dt + ∂y * dt

Question 9

Directional Derivative