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## Multivariable calculus

### Course: Multivariable calculus>Unit 2

Lesson 5: Multivariable chain rule

# Multivariable chain rule

This is the simplest case of taking the derivative of a composition involving multivariable functions. Created by Grant Sanderson.

## Want to join the conversation?

• Is this the 3Blue1Brown guy? Sounds really really similar! • Just want to clarify that this IS the Total Differential? I thought of this as instead of Multivariable Chain Rule, but product rule instead (since chain rule usually implied). Is that a different, but acceptable understanding of it? • Saul has introduced the multivariable chain rule by finding the derivative of a simple multivariable function by applying the single variable chain and product rules. He then rewrites the formula he has used in a manner equivalent to the multivariable chain rule to demonstrate that the multivariable chain rule is equivalent to applying rules that we already know to work.
• I'm surprised by how much the dot product comes up very often in multivar calc. Your essence of linear algebra series was really helpful! • grant is baccc • dx/dx =1 But dx/∂x= ? • In calculus, "dx" represents an infinitesimal change in the variable "x," and it's often used in the context of finding derivatives. When you write "dx/dx = 1," it means the derivative of "x" with respect to "x" is equal to 1, which is a tautological statement. Essentially, it's saying that a change in "x" with respect to "x" is always 1, which is true because it's a straightforward change in the same variable.

However, when you write "dx/∂x," you are taking the derivative with respect to a partial derivative (∂x), which typically implies that you are dealing with a function of multiple variables. The partial derivative symbol (∂) is used in multivariable calculus to indicate that you are taking a derivative with respect to one variable while keeping other variables constant.

So, "dx/∂x" doesn't have a straightforward interpretation without context. The result would depend on the specific function you are differentiating with respect to "x" (∂x) and how it depends on other variables.

In general, "dx/∂x" is a notation that isn't commonly used because it's somewhat ambiguous. You would typically see "∂f/∂x" to represent the partial derivative of a function "f" with respect to "x."

Let's consider a simple example of a function of two variables, say, "f(x, y) = x^2 + 2xy + y^2." We can find the partial derivative of this function with respect to "x," denoted as ∂f/∂x:

f(x, y) = x^2 + 2xy + y^2

∂f/∂x is found by treating "y" as a constant and taking the derivative of "f" with respect to "x." The derivative of "x^2" with respect to "x" is "2x," the derivative of "2xy" with respect to "x" is "2y," and the derivative of "y^2" with respect to "x" is 0 because "y" is a constant with respect to "x." So, we have:

∂f/∂x = 2x + 2y

Now, if you want to find "dx/∂x" for this function, you are essentially calculating the reciprocal of ∂f/∂x because "dx" is a small change in "x" and ∂f/∂x represents how "f" changes concerning "x." Therefore:

dx/∂x = 1 / (2x + 2y)

This gives you a sense of how "x" changes concerning the change in "x" (dx) for the given function, taking into account how it depends on both "x" and "y."

So, if you were to evaluate this expression for specific values of "x" and "y," you would find the rate of change of "x" concerning "x" for that point in the function.
(1 vote)
• why f[x(t),y(t)] is considered function of 1 variable ? • Grant is backkk! • To visualize f(x(t), y(t)) in 3D space, would t be the length of the curve?   