14.1 – Differentiability and Gradient

14.1 – Differentiability and Gradient We say that f is differentiable at x if there exists a vector y such that f (x + h) − f (x) = y ∙ h + o(h).

We will say that g(h) is o(h) if 0

( )lim 0h

g→

=h

h Example: For 2( , ) 3f x y x y= + : Let f be differentiable at x. The gradient of f at x is the unique vector ∇ f (x) such that f (x + h) − f (x) = ∇ f (x) ∙ h + o(h). Continuing the previous example:

NOTE: The gradient is a VECTOR!!!! More examples: 1. Find the gradient of 2( , ) 2 sin( )xf x y e x y= + 2. Find the gradient of ( , , )f x y z xy z= − at (2, -1, 4)

15.2 – Gradients and Directional Derivatives Properties of gradients:

Directional Derivatives:

u'f gives the directional derivative of f in the direction u. In other words, uf ′ gives the rate of change of f in the direction of u.

𝑓𝑓𝐮𝐮′(𝐱𝐱) = ∇𝑓𝑓(𝐱𝐱) ∙ 𝐮𝐮 Example:

1. Find the directional derivative at the point P in the direction indicated.

2 2( , ) 3 at P(1,1) in the direction of f x y x y= + i – j

2. Find the directional derivative for 2 2( , ) 3f x y x y= + at 11,2

Q −

towards 32,2

R

.

Note that the directional derivative in a direction u is the component of the gradient vector in that direction.

Important:

Example: Find a unit vector in the direction in which f increases most rapidly at P and give the rate of change of f in that direction; find a unit vector in the direction in which f decreases most rapidly at P and give the rate of change of f in that direction.

2 2( , ) e at P(0,1)xf x y y=

15.3 – The Mean Value Theorem; Chain Rules What was the MVT for functions of one variable?

Example: Let 3( , )f x y x xy= − and let a = (0,1) and b = (1,3). Find a point c on the line segment ab for which the mean value theorem (for several variables) is satisfied.

Examples:

1. Find [ ]( ( ))d f tdt

r given f (x, y) = 6x + y, r(t) = t i + 7t j

2. Find the rate of change of f with respect to t along the given curve. f (x, y) = x2y, r(t) = e t i + e−t j

3. Find the rate of change of f with respect to t along the given curve.

Other chain rules: If

Then

Example:

4. 2 23 2 ( ) cos ( ) sinu x xy y x t t y t t= − + = = . Find dudt

And if

Then

Example:

5. 2 2 cos sin( ) sinu x xy z x s t y t s z t s= − + = = − = . Find dudt