5.1 The chain rule for functions of several variables
3 min read•august 6, 2024
The chain rule for functions of several variables builds on what you already know about derivatives. It's like leveling up your calculus skills, allowing you to handle more complex situations involving multiple variables.
This rule is super useful when dealing with . It helps you break down tricky problems into manageable pieces, making it easier to find derivatives and understand how different variables affect each other.
Derivatives of Composite Functions
The Chain Rule and Composite Functions
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The chain rule extends the concept of the single-variable chain rule to functions of several variables
Enables finding the derivative of a composite function by multiplying the of the outer function with the of the inner function
A composite function is a function that depends on another function
For example, if f(x,y)=x2+y2 and g(t)=(t,t3), then f(g(t))=t2+t6 is a composite function
To apply the chain rule, express the composite function as a composition of an outer function and an inner function
The outer function takes the output of the inner function as its input
The inner function is a function of the original independent variables
Partial Derivatives and the Total Derivative
Partial derivatives measure the rate of change of a function with respect to one variable while holding other variables constant
For a function f(x,y), the partial derivative with respect to x is denoted as ∂x∂f and the partial derivative with respect to y is denoted as ∂y∂f
The is the generalization of the single-variable derivative to functions of several variables
It measures the rate of change of a function in any direction
For a function f(x,y), the total derivative is given by df=∂x∂fdx+∂y∂fdy
When applying the chain rule, the total derivative of the composite function is the product of the partial derivatives of the outer function and the total derivatives of the inner functions
For example, if f(x,y)=x2+y2 and x=g(t)=t and y=h(t)=t3, then dtdf=∂x∂fdtdx+∂y∂fdtdy=2xdtdx+2ydtdy=2t(1)+2t3(3t2)=2t+6t5
Multivariable Calculus Fundamentals
Multivariable Functions and Differentiability
Multivariable functions are functions that depend on multiple variables
For example, f(x,y)=x2+y2 is a function of two variables, x and y
A multivariable function is differentiable at a point if its partial derivatives exist and are continuous at that point
Differentiability ensures that the function is smooth and well-behaved near the point
Differentiability allows for the approximation of a function using a linear function (its total derivative) near a point
This linear approximation is the basis for many applications, such as optimization and error estimation
Gradient Vector and Jacobian Matrix
The of a multivariable function f(x1,x2,…,xn) is a vector of its partial derivatives
It points in the direction of the greatest rate of increase of the function
The gradient vector is denoted as ∇f=(∂x1∂f,∂x2∂f,…,∂xn∂f)
The is a matrix of partial derivatives for a vector-valued function f(x1,x2,…,xn)=(f1(x1,x2,…,xn),f2(x1,x2,…,xn),…,fm(x1,x2,…,xn))
It generalizes the gradient vector to vector-valued functions
The Jacobian matrix is denoted as Jf=∂x1∂f1∂x1∂f2⋮∂x1∂fm∂x2∂f1∂x2∂f2⋮∂x2∂fm⋯⋯⋱⋯∂xn∂f1∂xn∂f2⋮∂xn∂fm
The gradient vector and Jacobian matrix play crucial roles in optimization, coordinate transformations, and analyzing the behavior of multivariable functions