# Can the 3rd Derivative Theorem Simplify Calculating Higher Order Derivatives?

• Orion1
In summary, the conversation discusses the existence of a theorem for deriving second or third derivatives without having to first derive the lower order derivatives. The conversation also mentions the use of Taylor's theorem and the n! method for obtaining derivatives. However, it is concluded that these methods are not theorems and are just patterns of derivatives that can be manipulated.
Orion1

Does a theorem exist for the derivation of a second or third derivative equation, without having to first derive the first and second equation derivatives?

Example equation:
$$\frac{d^3}{dx^3} \left( \frac{x}{2x - 1} \right)$$

Orion1 said:

Does a theorem exist for the derivation of a second or third derivative equation, without having to first derive the first and second equation derivatives?

Example equation:
$$\frac{d^3}{dx^3} \left( \frac{x}{2x - 1} \right)$$

Take a look at Leibniz Identity.

Thats not what he's asking about, and to the OP: I don't believe so.

Hmm, yeah - if there is - it will make life much easier :p

Well, we can easily make them couldn't we :)?

$$\begin{array}{rcl}\frac{d^3}{dx^3}C &=& \frac{d}{dx}\frac{d}{dx}\frac{d}{dx}C\\ &=& \frac{d}{dx}\frac{d}{dx}0\\ &=& \frac{d}{dx}0\\ &=& 0\end{array}$$

Thus we get our first theorem:

$$\frac{d^3}{dx^3}C &=& 0$$

etc etc...

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C is a function of x.. you don't get a 0

I think he was being sarcastic and claiming C to stand for constant.

Nope,there's no theorem,but there are functions,"nice" ones,for which u can find,by a mere substitution,the derivative of arbitrary order.

To give you an example:take "sine".Compute its "n"-th order derivative.

Daniel.

Taylor's theorem.

If you can find the Taylor Series for the function around a, then you can read off the value of any derivative you want at a.

(Say... by using the geometric series formula)

Hurkyl's advice is very satisfactory...with the help of taylor series any derivative cn be found..

actually the example he gave is rather trivial if rewritten as [1/2][ 1 minus something like (2x-1)^(-1)]

n factorial...

My calculus book describes using an $$n!$$ method for obtaining derivatives:

Example equation:
$$y = \frac{1}{3x^3}$$

Where:
$$\frac{d^{n}y}{dx^{n}} = \frac{(-1)^n (n+2)!}{(6x^{n+3})}$$

Then:
$$\boxed{\frac{d^{3}y}{dx^{3}} = - \frac{20}{x^6}}$$

Unfortunately, the $$n!$$ method originates after obtaining several of the original derivatives first.

Example equation 2 (post#1):
$$y = \frac{x}{2x - 1}$$

Where:
$$\frac{d^{n}y}{dx^{n}} = \frac{(-1)^n (2^{n-1}) n!}{(2x-1)^{n+1}}$$

Then:
$$\boxed{\frac{d^{3}y}{dx^{3}} = - \frac{24}{(2x - 1)^4}}$$

Which seems to indicate the existence of a missing theorem, therefore, what is the $$n!$$ theorem?

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Theres no theorem here, you're just finding patterns of derivatives and manipulating them. This is taught in Calc I..

## 1. What is the 3rd Derivative Theorem?

The 3rd Derivative Theorem is a mathematical principle that states the third derivative of a function can be used to determine the concavity of the original function.

## 2. How is the 3rd Derivative Theorem used in real-world applications?

The 3rd Derivative Theorem is commonly used in physics, engineering, and economics to analyze the behavior of systems and make predictions. For example, it can be used to predict the stability of a structure or the growth rate of a population.

## 3. Can the 3rd Derivative Theorem be applied to all types of functions?

Yes, the 3rd Derivative Theorem can be applied to any continuous function. However, it is most commonly used for polynomial functions.

## 4. What is the relationship between the 2nd and 3rd derivatives in the 3rd Derivative Theorem?

The 3rd Derivative Theorem states that the second derivative of a function determines the concavity of the function, while the third derivative determines the rate of change of the second derivative. In other words, the 3rd derivative is the derivative of the second derivative.

## 5. Are there any limitations to the 3rd Derivative Theorem?

While the 3rd Derivative Theorem is a useful tool for analyzing functions, it does have limitations. It may not be applicable to all types of functions, and it does not provide information about other important characteristics of a function, such as its domain or range.

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