# Derivative of a function of another function

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• kent davidge
In summary: I'll try to be more clear in the future.Indeed... As if it wasn't enough not understanding the topic, I come up with this notation to make things more difficult. I'll try to be more clear in the future.
kent davidge
This is really a simple question, but I'm stuck.

Suppose we have a function ##\vartheta'(\vartheta) = \vartheta## and that ##\vartheta = \vartheta(\varphi)## and we know what ##\vartheta(\varphi)## is. How should I view ##\frac{\partial \vartheta'}{\partial \varphi}##? Should I set it equal to zero because ##\varphi## does not appear explicitely? But as I said, I know what ##\vartheta(\varphi)## is in this particular case, and it is not even linear in ##\varphi##, so it doesn't make sense to say that its derivatives vanish.

Edit: I added a crucial correction to my post.

What does ## \vartheta '( \vartheta )## mean?

fresh_42 said:
What does ## \vartheta '( \vartheta )## mean?
I'm sorry, that was a bad choice of notation. ##\vartheta'## is simply a function, the prime is there to differentiate it from ##\vartheta##. It does not mean derivative. If you prefer, let's call it ##\Theta## from now on to cause no confusion.

Still truly bad notation, especially because a prime superscript placed on the symbol for a function typically means its derivative. And this question is about derivatives.

fresh_42
zinq said:
Still truly bad notation, especially because a prime superscript placed on the symbol for a function typically means its derivative. And this question is about derivatives.

So we are talking about the chain rule? ##g=g(\varphi)## and ##f=f(g)##, so ##f=f(g(\varphi))##?

fresh_42 said:
So we are talking about the chain rule? ##g=g(\varphi)## and ##f=f(g)##, so ##f=f(g(\varphi))##?
oh yes. And I would like to know $$\frac{\partial f}{\partial \varphi}$$ If ##\varphi## were the argument of ##f## directly, then it would be the same as the total derivative of ##f## wrt it. But since ##\varphi## is the argument of ##g##, I'm not sure how to proceed.

chain rule?
oops, this just reminded me that we can use the chain rule also with partial derivatives. so $$\frac{\partial f}{\partial \varphi} = \frac{\partial f}{\partial g}\frac{\partial g}{\partial \varphi}$$

##\dfrac{df}{d \varphi}=\dfrac{df(g)}{dg}\cdot\dfrac{dg(\varphi)}{d\varphi}##.

Let's try different notation for the OP and use an example.

Suppose we are given ##f(y) = 2y## and ## y = x^2##. Then what is ##\frac {df}{dx}##?

Is it zero because ##x## does not appear in the formula ##f(y) = 2y## ? Or is it ##4x## because ##f(x) = 2(x^2)## ?

This involves ambiguous and bad notation - and notation that is customary, especially in writing about physics! Beginning students in mathematics are taught that a function has a definite domain and co-domain. So if two functions have different co-domains, they must be different functions. The function ##f(y)## can take on negative values. The function ##f(x)## cannot. So we shouldn't use the name ##f## for both functions. However, it is common practice to do so.

When we are "given" that ##y = x^2##, it isn't clear how the name ##f## should be used in representing this fact. To be unambiguous, we could say there is function ##f(y) = 2y## and another function ##B(x) = f(x^2) = 2x^2##.

Technically speaking, if ##x## denotes a variable not in the domain of ##f## then ##\frac{df}{dx}## is undefined. However, convention says that we interpret ##\frac{df}{dx}## by pretending ##f(y)## is actually a function of two variables ##(x,y)## that is constant with respect to ##x##. With that convention, ##\frac{df}{dx} = \frac{\partial f}{\partial_x} = 0##.

I'd say the most common interpretation of the above example in physics is that ##\frac{df}{dx} = 0##.

In a math text, the author might want to give students an exercise in the chain rule. He would expect them to interpret ##\frac{df}{dx}## as the derivative of the function ##B(x)##.

kent davidge
kent davidge said:
Suppose we have a function ##\vartheta'(\vartheta) = \vartheta## and that ##\vartheta = \vartheta(\varphi)## and we know what ##\vartheta(\varphi)## is.
Why are you using Greek letters here? The examples shown in the posts by @fresh_42 and @Stephen Tashi are much clearer, not to mention easier to type.

kent davidge
Mark44 said:
Why are you using Greek letters here? The examples shown in the posts by @fresh_42 and @Stephen Tashi are much clearer, not to mention easier to type.
Indeed... As if it wasn't enough not understanding the topic, I come up with this notation to make things more difficult.

## 1. What is the derivative of a function of another function?

The derivative of a function of another function, also known as a composite function, is the rate of change of the outer function with respect to the inner function. It is found by using the chain rule, which states that the derivative of a composite function is equal to the derivative of the outer function multiplied by the derivative of the inner function.

## 2. How do you find the derivative of a function of another function?

To find the derivative of a function of another function, you first need to identify the inner and outer functions. Then, use the chain rule to take the derivative of the outer function and multiply it by the derivative of the inner function. Make sure to also apply the chain rule to the inner function if it is also a composite function.

## 3. Can you give an example of finding the derivative of a function of another function?

Sure, let's say we have the function f(x) = (x^2 + 1)^3. The inner function is x^2 + 1 and the outer function is ( )^3. Using the chain rule, the derivative would be f'(x) = 3(x^2 + 1)^2 * 2x = 6x(x^2 + 1)^2.

## 4. What is the purpose of finding the derivative of a function of another function?

The derivative of a function of another function is useful in calculating the rate of change of a variable in a complex system. It can also help in finding the maximum and minimum values of a function, as well as determining the slope of a tangent line at a specific point on the graph of the function.

## 5. Are there any other methods for finding the derivative of a function of another function?

Yes, there are other methods such as the product rule, quotient rule, and power rule that can be used to find the derivative of a function of another function. However, the chain rule is the most commonly used and efficient method for this type of derivative.

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