Master the Chain Rule: Derivative of f(x) = \sqrt{5-x^{2}} and Its Composition

[Incog]

Homework Statement

A: Write f(x) = $$\sqrt{5-x^{2}}$$ as a composite of two functions.

B: Use the Chain Rule to find the derivative of f(x) = $$\sqrt{5-x^{2}}$$

Homework Equations

Chain Rule:

y`= $$\frac{dy}{du}$$ $$\frac{du}{dx}$$

The Attempt at a Solution

A:

y = $$\sqrt{u}$$
u = 5 - x$$^{2}$$


B:

This is where I get confused. I don't understand what's meant by "d" and what's meant by "y", "u", and "x".

I know the two du's cancel out in the Chain Rule so you're left with:

y`= (dy)(dx)

Does this mean the derivative of y times the derivative of x? And if so, how do you know what y and x are?

 

[Doc Al]

y = $$\sqrt{u}$$
u = 5 - x$$^{2}$$

Good!

This is where I get confused. I don't understand what's meant by "d" and what's meant by "y", "u", and "x".

dy/dx is just notation for: take the derivative of the function y with respect to the variable x.

I know the two du's cancel out in the Chain Rule so you're left with:

y`= (dy)(dx)

Not exactly. y' = dy/dx.

The point is that you want to find dy/dx and that dy/dx = (dy/du)*(du/dx). And it's easy to calculate the derivatives dy/du and du/dx.

 

[Incog]

dy/dx is just notation for: take the derivative of the function y with respect to the variable x.

Thanks. That cleared the confusion.