# Taking 2nd, 3rd and 4th derivatives

• Cacophony
In summary, the second derivative of ln(x+1) is -(1/(x+1)^2), the third derivative of x^7 + 4x^6 - x^2 is 210x^4 + 480x^3, the second derivative of 1/(x+1) is 2/(x+1)^3, and the fourth derivative of cos(2x) is 16cos(2x).
Cacophony

## Homework Statement

a) d^2/(dx^2): ln(x+1)
b) d^3/(dx^3): x^7 + 4x^6 - x^2
c) d^2/(dx^2): 1/(x + 1)
d) d^4/(dx^4): cos(2x)

none

## The Attempt at a Solution

Can someone tell me if these are right?

a)= -(1/(x + 1)^2)
b)=210x^4 + 480x^3
c)= (2/(x + 1)^3)
d)= 16cos(2x)

Hi Cacophony!

(try using the X2 button just above the Reply box )
Cacophony said:

## Homework Statement

a) d^2/(dx^2): ln(x+1)
b) d^3/(dx^3): x^7 + 4x^6 - x^2
c) d^2/(dx^2): 1/(x + 1)
d) d^4/(dx^4): cos(2x)

none

## The Attempt at a Solution

Can someone tell me if these are right?

a)= -(1/(x + 1)^2)
b)=210x^4 + 480x^3
c)= (2/(x + 1)^3)
d)= 16cos(2x)

Yes, they're fine.

## What is the purpose of taking 2nd, 3rd, and 4th derivatives?

Taking higher order derivatives allows us to analyze the rate of change and curvature of a function at a given point. This can provide insights into the behavior and properties of the function.

## What is the process for taking a 2nd, 3rd, or 4th derivative?

To take a higher order derivative, we follow the same steps as taking a first derivative, but repeat the process multiple times. This involves applying the power rule, product rule, quotient rule, or chain rule as necessary.

## How do higher order derivatives relate to each other?

The 2nd derivative represents the rate of change of the 1st derivative, the 3rd derivative represents the rate of change of the 2nd derivative, and so on. In other words, each higher order derivative provides information about the rate of change of the derivative before it.

## What can we learn from the 2nd, 3rd, and 4th derivatives of a function?

The 2nd derivative can tell us about the concavity of a function, the 3rd derivative can tell us about points of inflection, and the 4th derivative can provide information about the curvature of the function. All of these can give us a better understanding of the behavior of the function.

## Are there any practical applications of taking higher order derivatives?

Yes, higher order derivatives are used in fields such as physics, engineering, and economics to analyze and model real-world phenomena. For example, the 2nd derivative can be used to calculate acceleration, and the 3rd derivative can be used to analyze the stability of a system.

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