Second Derivative of e^t + t^e ?

In summary, to find the second derivative of e^t + t^e, we use the rule (e^u) = u' * (e^u) and first find the first derivative which is (e^t) + e*[t^(e-1)]. Then, we apply the rule again to find the second derivative, taking into account the change in the exponent of t, resulting in (e^t) + e^2*[t^(e-2)] as the final answer. It is important to note that the derivative of e^x is e^x and not just e.
  • #1
catteyes
9
0

Homework Statement



Find the second derivative of e^t + t^e :


Homework Equations



(e^u) = u' * (e^u)


The Attempt at a Solution



e^t + t^e

1* (e^t) + e*[t^(e-1)]

^^^first derivative

(e^t) + e*e[t^{(e-1)-1}]

^^^ 2nd derivative

Answer?: (e^t) + e^2[t^(e-2)]
 
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  • #2
catteyes said:
Answer?: (e^t) + e^2[t^(e-2)]

you're quite close. after taking the first derivative, the exponent of t changes, you have not accounted for that.
 
  • #3
I'm not sure I follow. Doesn't the exponent of t become (e-2) ?
 
  • #4
d/dt (t^e) = e*t^(e-1)

d/dt (e*t^(e-1)) = ??

EDIT/HINT: Your exponent is correct, it is the constant in front of the exponent that isn't. It's quite a common error, though you probably won't be making it in the future.
 
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  • #5
scratch that... it didn't even look almost right
 
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  • #6
catteyes said:
e^t + t^e

1* (e^t) + e*[t^(e-1)]

[tex]t^e[/tex]
[tex]t^u = u' f'(u)[/tex]

What is u'?
 
  • #7
u' = e

?

:/
 
  • #8
catteyes said:
u' = e

?

:/

Wrong, derivative of e^x is e^x; don't confuse that with "e"

hint: e itself is equal to what?
 

FAQ: Second Derivative of e^t + t^e ?

What is the second derivative of et + te?

The second derivative of et + te is et + 2et + t2et.

How do you find the second derivative of et + te?

The second derivative of et + te can be found by taking the derivative of the first derivative. This means finding the derivative of et and te separately, and then adding them together.

What is the significance of the second derivative in relation to et + te?

The second derivative of et + te represents the rate of change of the first derivative, which in turn represents the rate of change of the original function. This can be useful in understanding the behavior of the function and identifying points of inflection or maximum/minimum values.

How does the second derivative of et + te relate to other mathematical concepts?

The second derivative of et + te is closely related to the concept of concavity, which describes the shape of a function's graph. A positive second derivative indicates a concave up shape, while a negative second derivative indicates a concave down shape. Additionally, the second derivative is used in the process of optimization, where the maximum or minimum values of a function are determined.

What are some real-world applications of the second derivative of et + te?

The second derivative of et + te can be used in various fields such as physics, economics, and engineering. In physics, it can be used to analyze the motion of objects, while in economics it can help with predicting market trends. In engineering, the second derivative can be used to optimize designs and improve efficiency.

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