leibniz notation and chain rule

[Victoria_k_tori]

I need some help on a homework question i don't even know where to start, "Using the Leibniz notation, apply the chain rule to differentiate, dy/dx, y=√(u) - (2u), u=3x^2-1"

 

[Dr.Peterson]

I need some help on a homework question i don't even know where to start, "Using the Leibniz notation, apply the chain rule to differentiate, dy/dx, y=√(u) - (2u), u=3x^2-1"

Can you find an example in your textbook or notes, and show that to us? That's a good place to start when you don't know where to start: with what you were taught.

If you can't find that, I searched for "leibniz notation chain rule", and found this textbook that covers the topic (near the bottom):

3.6: The Chain Rule

Key Concepts The chain rule allows us to differentiate compositions of two or more functions. It states that for \(h(x)=f(g(x)),\) \(h′(x)=f′(g(x))g′(x).\) We can use the chain rule …

math.libretexts.org

Once you are able to start (even if you have no idea whether you are right), please show your attempt, so we can correct you if necessary.

 

[JeffM]

Leibniz thought of derivatives as fractions and devised his notation in light of that thought. The chain rule is amazingly intuitive in his notation.

[MATH]\dfrac{dy}{dx} = \dfrac{dy}{\cancel {du}} * \dfrac{\cancel {du}}{dx}[/MATH].

Now a modern mathematician will tell you that the cancellation is nowhere close to rigorous because derivatives are limits rather than fractions, but the Leibniz notation does make it easy to remember the chain rule.

 

[HallsofIvy]

\(\displaystyle y= \sqrt{u}- 2u= u^{1/2}- 2u\) and \(\displaystyle u= 3x^2- 1\).

What is \(\displaystyle \frac{dy}{du}\)? What is \(\displaystyle \frac{du}{dx}\)?