The Chain Rule and Directional Derivatives, and the Gradient Functions of Two Variables

Tutoring on The Chain Rule and Directional Derivatives, and the Gradient Functions of Two Variables

Learning Objectives:

Understand and Apply The Chain Rule and Directional Derivativesand the Gradient Functions of Two Variables.

Math Tutoring on The Chain Rule and Directional Derivatives, and the Gradient Functions of Two Variables

The Chain Rule: Functions with One Independent Variable:

If z = f(x, y) and x = g(t) and y = h(t) with z, x and y differentiable functions then,

Functions with Two Independent Variables:

If z = f(x, y) and x = g(s, t) and y = h(s, t) with z, x and y differentiable functions of s and t, then

Two Independent Variables s and t.

Directional Derivatives of Two Variables:

Let f be a function of two variables x and y and let  be a unit vector. Then the Directional Derivative of f in the direction of u, denoted by  is

Provided the Limit Exists.

Gradient Functions of Two Variables:

In math tutoring, we can also use the notation grad(f). We can say gradient of f or del f”. The Directional Derivative of f is equal to the gradient of f(x, y) dotted with the unit vector u.

 

 

 

 

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Hook Questions:

1.     Describe the Chain Rule of differentiation for the functions with two variables?

2.     State Directional Derivatives of two variables?

3.     State Gradient functions of two variables?

 

 

 

 

 

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