First and Second Derivatives

  • Thread starter mmajames
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In summary, the conversation was about finding the first and second derivatives of various functions using the Quotient Rule, Power Rule, and Chain Rule. The student was having trouble with the second derivatives and shared their attempts at solving them. The expert summarized the steps for finding the derivatives and confirmed that the student's solutions were correct.
  • #1
mmajames
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Homework Statement


Just trying to find the first and second derivatives.

X^2/(X^2-16)

1+X/1-X

X^3(X-2)^2

Homework Equations


Quotient Rule/Power Rule/Chain Rule


The Attempt at a Solution

 
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  • #2
For the first one, what is the derivative of f(x)/g(x)? In general, not for this specific function.
 
  • #3
(g(x)*f'(x)-f(x)*g'(x))/g(x)^2
 
  • #4
okay, so if f(x) = x^2 and g(x) = x^-2, what is the derivative of f(x)/g(x)?
 
  • #5
f(x) = x^2 and g(x) = x^-2
x^2/x^-2
((x^-2)(2x)-(-2x)(x^2))/ (x^2)^2
 
  • #6
I mistyped (that g(x) should have been g(x)=x^2 - 16 [not sure how I messed that one up]), but you seem to know what you're doing. What specific question do you have?
 
  • #7
I'm mostly having trouble with the second derivatives.
 
  • #8
What do you have so far? Where are you getting stuck?
 
  • #9
Well here's what I've got, I think they're right but I'm not sure.

f(x)=X^2/(x^2-16)
(X^2-16)(2X)-(X^2)(2X)/(X^2-16)^2
f'(x)=-32X^2/(X^2-16)^2
(X^4-32X^2+256)(-64X)-(-32X^2)(4X^3-64X)/(X^4-32X^2+256)^2
-64x^5+2048X^3-16384X+128X^5-2048X^3
f''(x)=64X^5-16384X/(X^4-32X^2+256)^2

f(x)=1+x/1-X
(1-X)(1)-(1+X)(-1)/(1-X)^2
f'(x)=2-2X/(1-X)^2
(X^2-2X+1)(-2)-(2X-2)(2-2X)/(X^2-2X+1)^2
f''(x)=2X^2+4X+2/(X^2-2X+1)^2

f(x)=X^3(X-2)^2

(X^3)(2X-4)+(3X^2)(X-2)^2
2X^4-4X^3+3X^4-12X^3+12X^2
f'(x)=5X^4-16X^3+12X^2
5X^4-16^3+12X^2
f''(x)=20X^3-48X^2+24X
 
  • #10
Am I doing these correctly?
 

1. What is the difference between first and second derivatives?

The first derivative of a function represents its rate of change at a specific point, while the second derivative represents the rate at which the first derivative changes. In other words, the first derivative tells us how fast a function is changing, and the second derivative tells us how fast the rate of change is changing.

2. How are first and second derivatives useful in real life?

First and second derivatives are used in many fields, such as physics, economics, and engineering, to analyze and model various phenomena. For example, the first derivative can be used to determine the velocity of an object at a specific time, and the second derivative can be used to determine the acceleration of the object.

3. What is the process for finding the first derivative of a function?

The first derivative of a function can be found by taking the derivative of the function using the rules of differentiation. These rules include the power rule, product rule, quotient rule, and chain rule. The resulting derivative will be a new function that represents the rate of change of the original function.

4. How can the second derivative be used to find critical points of a function?

The second derivative can be used to find the critical points of a function, which are the points where the function changes from increasing to decreasing or vice versa. These points can be found by setting the second derivative equal to zero and solving for the variables. If the resulting values fall within the domain of the function, they are considered critical points.

5. Can first and second derivatives be used to determine the concavity of a function?

Yes, the first and second derivatives can be used to determine the concavity of a function. The concavity of a function indicates whether the graph of the function is curving upwards or downwards. A positive second derivative indicates a concave up function, while a negative second derivative indicates a concave down function.

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