# Nth Order Derivative of f(x): Sin^4(x)+Cos^4(x) & x^n/(1-x)

• shrody
In summary, the student is trying to express sin^4x and cos^4x in terms of multiple angles without using powers, and then taking the derivatives of those. The first fraction can be expanded into -(1+x+x^2+...+x^{n-1}) and the second fraction can be divided by x-1 to get the long expansion thing.
shrody

## Homework Statement

The exercise goes "Determine d^n*f/dx^n for the 2 exercises:"
a) f(x)=sin^4(x) + cos^4(x)
b) f(x)= x^n/(1-x)

## The Attempt at a Solution

The only idea i had was for the second example, where i think its right to use a rule from Leibniz but I'm not sure...

I've never done any of these before, so please forgive me if I'm way off and not being helpful at all.

a) couldn't you express $sin^4x$ and $cos^4x$ in terms of multiple angles without powers, and then take the derivatives of those?

b) you can make the fraction like this: $$\frac{-(1-x^n)+1}{1-x}$$

and then they split up as so: $$\frac{-(1-x^n)}{1-x}+(1-x)^{-1}$$

The first fraction can be... expanded (for lack of a better word) into $-(1+x+x^2+...+x^{n-1})$ and I'm sure it's clear what to do with the second fraction.

i don't know about the first one with the angles, i think the whole point of the exercise is to keep the sin and cos and the second one kinda confused me...

Well you would still keep the sin and cos, just that theyre expressed as

$$sin^nx=Asin(nx)+Bsin((n-1)x)...$$

I know you could find the relationship easily for the n-th derivative from that, but if that isn't allowed, then maybe this will help?

$$y=sin^4x$$

$$\frac{dy}{dx}=4sin^3xcosx$$

$$\frac{d^2y}{dx^2}=-16sin^4x+12sin^2x$$

Notice how the $sin^4x$ appears again. Maybe it's something, probably it's not, but I'm just putting that out there in case it helps.

For the second, is that "confused me" or "still confuses me"?

$$(x^n-1)=(x-1)(x^{n-1}+x^{n-2}+...+x+1)$$

dividing both sides by $x-1$ will show you how to get the long expansion thing.

## 1. What is the formula for the Nth order derivative of f(x)?

The formula for the Nth order derivative of f(x) is given by taking the derivative of the function n times. For example, the first derivative is f'(x), the second derivative is f''(x), and so on. In general, the Nth order derivative is denoted by f(N)(x).

## 2. How do you find the Nth order derivative of a trigonometric function?

To find the Nth order derivative of a trigonometric function, such as sin4(x) + cos4(x), we use the power rule and chain rule. First, we expand the function using the binomial theorem and then use the power rule to find the derivative of each term. Then, we use the chain rule to find the derivative of the entire function. This process is repeated n times for the Nth order derivative.

## 3. What is the Nth order derivative of a polynomial function?

The Nth order derivative of a polynomial function, such as xn/(1-x), can be found using the power rule and product rule. We first expand the function using the binomial theorem and then use the power rule to find the derivative of each term. Then, we use the product rule to find the derivative of the entire function. This process is repeated n times for the Nth order derivative.

## 4. Can the Nth order derivative of a function be negative?

Yes, the Nth order derivative of a function can be negative. The sign of the Nth order derivative depends on the function and the value of n. For example, if n is an even number, the Nth order derivative of a function may be positive for some values of x and negative for others.

## 5. What is the relationship between the Nth order derivative and the original function?

The Nth order derivative of a function represents the rate of change of the N-1th order derivative. In other words, the Nth order derivative is the derivative of the (N-1)th order derivative. This relationship can be seen in the power rule and chain rule used to find the Nth order derivative, where the derivative of each term is multiplied by the derivative of the entire function.

• Precalculus Mathematics Homework Help
Replies
11
Views
2K
• Precalculus Mathematics Homework Help
Replies
6
Views
1K
• Precalculus Mathematics Homework Help
Replies
3
Views
1K
• Precalculus Mathematics Homework Help
Replies
2
Views
1K
• Topology and Analysis
Replies
4
Views
688
• Precalculus Mathematics Homework Help
Replies
10
Views
4K
• Precalculus Mathematics Homework Help
Replies
2
Views
1K
• Precalculus Mathematics Homework Help
Replies
12
Views
1K
• Precalculus Mathematics Homework Help
Replies
2
Views
1K
• Calculus
Replies
12
Views
2K