How Do You Prove the Multivariable Chain Rule?

In summary, the multivariable chain rule is proven using the concept of differentiability and the properties of limits. By considering a composite function \( z = f(g(x, y), h(x, y)) \), the proof involves applying the single-variable chain rule to each component function while analyzing the partial derivatives with respect to each variable. The result is a formula that expresses the total derivative of the composite function in terms of the partial derivatives of the inner functions and the outer function, thus establishing a systematic approach to differentiating multivariable functions.
  • #1
Lambda96
223
75
Homework Statement
Use the chain rule to show the followin is true ##D(f+g)=D(f)+D(g)##
Relevant Equations
Multivariable Chain rule
Hi,

Im completly lost regarding the following exercise:

Bildschirmfoto 2024-06-14 um 20.43.19.png


Unfortunately, I don't understand how to prove the statement using the chain rule. The chain rule is always used if there is a composition, i.e. ##f\circ g=f(g(x))## then I first have to calculate ##g(x)## and insert this result into ##f##, but D(f+g) is a operation and not a composition.
 
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  • #2
Lambda96 said:
Homework Statement: Use the chain rule to show the followin is true ##D(f+g)=D(f)+D(g)##
Relevant Equations: Multivariable Chain rule

Hi,

Im completly lost regarding the following exercise:

View attachment 346915

Unfortunately, I don't understand how to prove the statement using the chain rule. The chain rule is always used if there is a composition, i.e. ##f\circ g=f(g(x))## then I first have to calculate ##g(x)## and insert this result into ##f##, but D(f+g) is a operation and not a composition.
The addition is the second function, say ##A(f,g)= f+g.## Then we get ##D(f+g)=D(A(f,g))=(D\circ A)(f,g)## and ##D(f)+D(g)= (D(A))(D(f),D(g)).##
 
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  • #3
Thank you fresh_42 for your help 👍, now the task makes more sense


If I now apply the chain rule to the expression ##(D \circ A)(f,g)##, the result should be ##D'(A(f,g))\cdot A'(f,g)##, right?
 
  • #4
I'm not sure how to write it best and it is all about notation.

With ##A(f,g)=f+g## we have
\begin{align*}
A(\alpha (f,g))&=A(\alpha f, \alpha g)=\alpha f+\alpha g=\alpha(f+g)=\alpha A(f,g)\\
A((f_1,g_1)+(f_2,g_2))&=A((f_1+f_2),(g_1+g_2))=(f_1+f_2)+(g_1+g_2)\\
&=(f_1+g_1)+(f_2+g_2)=A(f_1,g_1)+A(f_2,g_2)
\end{align*}
and ##A## is linear. Therefore we have ##DA=A.## Finally,
$$
D(f+g)=D(A(f,g))=DA(D(f,g))=DA(D(f),D(g))=A(D(f),D(g))=D(f)+D(g)
$$

Please note to every equation sign which property we have used!
1. ...
2. ...
3. ...
4. ...
5. ...
 
Last edited:
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  • #5
Many thanks for your help fresh_42 👍 now I have understood it :smile:
 

FAQ: How Do You Prove the Multivariable Chain Rule?

1. What is the multivariable chain rule?

The multivariable chain rule is a formula used to compute the derivative of a composite function when the function depends on multiple variables. It allows us to differentiate functions that are composed of other functions, taking into account how changes in the input variables affect the output.

2. How is the multivariable chain rule different from the single-variable chain rule?

The single-variable chain rule applies to functions of a single variable and involves the derivative of the outer function multiplied by the derivative of the inner function. In contrast, the multivariable chain rule involves partial derivatives and accounts for multiple input variables, requiring the use of the gradient vector to capture the effect of all variables simultaneously.

3. Can you provide a basic example of applying the multivariable chain rule?

Sure! Consider a function z = f(x, y) where x and y are functions of t, such that x = g(t) and y = h(t). To find dz/dt, we use the multivariable chain rule: dz/dt = (∂f/∂x)(dx/dt) + (∂f/∂y)(dy/dt). This expression combines the partial derivatives of f with respect to x and y, multiplied by the derivatives of x and y with respect to t.

4. What are the conditions for applying the multivariable chain rule?

The multivariable chain rule can be applied when the functions involved are differentiable, which means they have continuous partial derivatives in the relevant domains. Additionally, the composite function must be well-defined, meaning that the inner functions must produce valid outputs for the outer function.

5. How can I visualize the multivariable chain rule?

Visualization can be done using graphs of the functions involved. For instance, you can plot the surfaces of the functions f(x, y) and the paths defined by g(t) and h(t) in a three-dimensional space. The chain rule can be interpreted as describing how changes along the path defined by t affect the height of the surface defined by f, illustrating the relationship between the variables.

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