How Do You Calculate the Third Derivative of \( y = \frac{1}{x+1} \)?

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In summary, the third derivative of Y is the third order derivative with respect to x, denoted as d3y/dx3. To solve for it, you need to take the third derivative of Y using the chain rule. For Y = 1/X+1, d3y/dx3 can be simplified to -6/x^4. Solving for d3y/dx3 is significant in understanding the rate of change of the second derivative of Y, finding points of inflection, and approximating the curvature of the graph of Y. It is also the same as the third derivative of Y, with d3y/dx3 being the notation used in calculus and third derivative being the more common term in other fields
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mo7tn
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Homework Statement


d3y/dx3 OF Y= 1/X+1


Homework Equations



Y= 1/X+1

The Attempt at a Solution



i TRIED USING THE QUOTIENT RULE BUT BECAME STUCK AFTER DOING- (X+1)D/DX(1) + (1)D/DX(X+1)= (X+1)(1)+(1)(1)=1+1+1= 3 BUT HOW DO GET TO THE THRID DEGREE IM LOST
 
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  • #2
[tex]y=\frac{1}{x+1}[/tex]

[tex]\frac{dy}{dx}=\frac{1(x+1)-1(1)}{(x+1)^2}[/tex]

[tex]\frac{dy}{dx}==\frac{x}{(x+1)^2}[/tex]
then just differentiate that twice again and you'll get [itex]\frac{d^3y}{dx^3}[/itex]
 

FAQ: How Do You Calculate the Third Derivative of \( y = \frac{1}{x+1} \)?

What is the third derivative of Y?

The third derivative of Y is also known as the third order derivative or the third derivative with respect to x. It is denoted as d3y/dx3 and represents the rate of change of the second derivative of Y with respect to x.

How do you solve for d3y/dx3?

To solve for d3y/dx3, you will need to take the third derivative of Y. This can be done by taking the derivative of the second derivative of Y, which is d2y/dx2, using the chain rule. The resulting equation will be d3y/dx3 = d2y/dx2 * d/dx(1/x+1).

Can you simplify d3y/dx3 for Y= 1/X+1?

Yes, d3y/dx3 can be simplified for Y= 1/X+1. In this case, the first derivative of Y, dY/dx, is equal to -1/x^2. The second derivative, d2y/dx2, is equal to 2/x^3. Therefore, the third derivative, d3y/dx3, can be simplified to -6/x^4.

What is the significance of solving for d3y/dx3?

Solving for d3y/dx3 is important as it helps us understand the rate of change of the second derivative of Y with respect to x. It can also be used to find points of inflection or points where the concavity of the function changes, and to approximate the curvature of the graph of Y.

Is d3y/dx3 the same as the third derivative of Y?

Yes, d3y/dx3 is the same as the third derivative of Y. Both represent the rate of change of the second derivative of Y with respect to x. The notation d3y/dx3 is often used in calculus, while the term third derivative is more commonly used in other fields of science and engineering.

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