Understanding Derivatives: My Struggle

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In summary, the conversation involves a discussion on understanding the derivative of a function and how to use the Chain Rule to find it. The expert summarizer advises to focus on the current problem and not stress about future mathematics. They also give an example of using the Chain Rule to find the derivative of a function.
  • #1
superdave
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I don't understand it. How it works, etc...

I've read the book definition, looked it up on the internet, and I still don't get it.

Like the current problem I'm working on:

derivative of sin((cos x)^2)*cos((sin x)^2)
 
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  • #2
Read it again. Then try to identify 'what is what' in your problem.
 
  • #3
Here's the theory.

If f is a function, of x, what is the derivative of f (when/if it exists)?

It is another function df satisfying the property, that for small e

f(x+e)=f(x)+e*df(x) + o(e^2)

o(e^2) signifies the remainder, and it behaves quadratically, or worse, in e.

Now, it is easy to see what the derivative of f(g(x)) ought to be, if we are blase about things like e^2 and smaller terms. And there is no reason not to be so we can 'see' what is really going on.

If we add e to x, and if f and g are differentiable:
f(g(x+e) = f(g(x)+ e*dg(x) + o(e^2)) = f(g(x)) + e*dg(x)df(g(x)) + o(e^2).Now, look at your function. It is a product of two functions of functions. We can differentiate that just using all the rules we've learnt, so do so. Remember, the general case helps you understand what you're doing. The examples help you practice it.
 
  • #4
Well I got this:

cos(sin^2 x) * 2cosx *-sinx*cos(sin^2 x) + -sin(sin^2 x) * 2cosx * cosx *sin(cos^2 x)

the back of the book gives me -sin2x cos(cos2x)
 
  • #5
You know your trig identities, right? (though I think you might be a little out in your differentiation.)

sin(2x)=2sin(x)cos(x), and so on.
 
  • #6
of course I'm out on my differention. This textbook is written in the most obscure way possible. I missed the lecture where we went over this. My head really hurts. I'm malnurished because college dining halls care more about being cheap than healthy.

And I'm supposed to be a physics major. But if I can't understand this, how am I supposed to understand more complex math and physics?
 
  • #7
superdave said:
of course I'm out on my differention. This textbook is written in the most obscure way possible. I missed the lecture where we went over this. My head really hurts. I'm malnurished because college dining halls care more about being cheap than healthy.

And I'm supposed to be a physics major. But if I can't understand this, how am I supposed to understand more complex math and physics?

So, you can live without understanding the Chain Rule today. Worry about the mathematics you are doing now, for now, and worry about the mathematics of tomorrow, for tomorrow. You aren't there yet, so why stress about it

It basically goes like this...

h(x) = f(g(x))

So, h(x) is like a function with a function inside of it. See it? We have g(x) "inside" f(x).

What's the derivative of h(x)? Well, the textbook should say...

h'(x) = g'(x)*f'(g(x))

So, the derivative of h(x) is simply the derivative of "inside" function multiplied with the "derivative" of f(x) then we put g(x) back "inside" of f(x).

Note: I use quotes because it isn't formal. My intention is only to show you how to use it, and maybe later you will understand it.

Here is an example:

h(x) = (x+x^2)^2

So, your g(x) is the "inside" function, which is g(x) = x+x^2. Your f(n) is the outside function, which is f(n) = n^2, where n=(x+x^2)=g(x). Now, you understand, why I used n for f(n) instead of x here.

So, find the derivative of h(x).

Using the formula...

h'(x) = g'(x)*f'(g(x))

g'(x) = 1+2x *If you don't know this, you have bigger problems.
f'(n) = 2n

So, input that in the formula and we get...

h'(x) = (1+2x)*2n = (1+2x)*2*(x+x^2)

Note: g(x) = n

And, we are done.
 

Related to Understanding Derivatives: My Struggle

1. What are derivatives?

Derivatives are financial instruments that derive their value from an underlying asset, such as stocks, commodities, currencies, or interest rates. They allow investors to speculate on the future price movements of the underlying asset without actually owning it.

2. How do derivatives work?

Derivatives work by entering into a contract between two parties, a buyer and a seller, where they agree to exchange the underlying asset at a specific price and date in the future. This allows investors to profit from changes in the price of the underlying asset without actually buying or selling it.

3. What are the different types of derivatives?

There are several types of derivatives, including options, futures, forwards, and swaps. Options give the buyer the right, but not the obligation, to buy or sell the underlying asset at a set price and date. Futures and forwards are contracts to buy or sell the underlying asset at a predetermined price and date. Swaps involve exchanging cash flows based on different financial instruments.

4. What are the benefits of using derivatives?

Derivatives can offer several benefits, such as hedging against potential losses, speculating on future price movements, and diversifying investment portfolios. They also allow investors to take on leverage, meaning they can control a larger position with a smaller amount of capital.

5. What are the risks associated with derivatives?

While derivatives can offer benefits, they also come with risks. The main risk is that the underlying asset may not perform as expected, resulting in financial losses. Derivatives also involve complex financial instruments and can be highly leveraged, which can increase the risk of significant losses. It is important for investors to thoroughly understand the risks associated with derivatives before investing in them.

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