SOLUTION: if y = sin (t) , x = ln (t) , then ,(d ^2 y)/(dx ^2)=... when (x = π)

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Question 1208213: if y = sin (t) , x = ln (t) , then ,(d ^2 y)/(dx ^2)=... when (x = π)
Found 3 solutions by math_tutor2020, ikleyn, mccravyedwin:
Answer by math_tutor2020(3817)   (Show Source): You can put this solution on YOUR website!

x = ln(t)
t = e^x
y = sin(t)
y = sin(e^x)

First Derivative using Chain Rule.
y = sin(e^x)
dy/dx = cos(e^x)*d/dx[ e^x ]
dy/dx = cos(e^x)*e^x

Second Derivative using Product Rule and Chain Rule.
dy/dx = cos(e^x)*e^x
(d^2y)/(dx^2) = -sin(e^x)*e^x*e^x + cos(e^x)*e^x
(d^2y)/(dx^2) = e^x * ( cos(e^x) - sin(e^x)*e^x )

Evaluate at x = pi
(d^2y)/(dx^2) = e^x * ( cos(e^x) - sin(e^x)*e^x )
(d^2y)/(dx^2) = e^pi * ( cos(e^pi) - sin(e^pi)*e^pi )
(d^2y)/(dx^2) = 479.215377
The decimal value is approximate.
The calculator must be set to radian mode.
Answer by ikleyn(52915)   (Show Source): You can put this solution on YOUR website!
.
if y = sin (t) , x = ln (t) , then ,(d ^2 y)/(dx ^2)=... when (x = π)
~~~~~~~~~~~~~~~~~~~~~~~~ 

If  x = ln(t),  then  

    t = ,      (1)

where "e" is the base of natural logarithms.


Therefore, in this problem, after making substitution (1), we have

    y = sin(e^x),     (2)

i.e. function y is expressed as the composition of function sine and exponent.


So, we apply the formula for the derivative of a composite function and find 
first derivative of y with respect to x

     =  = .    (3)


Then we find second derivative as the derivative of (3)

    d^2 y
   ------- (x) =  +  =  + .
    d^2 x


Now we substitute x =  to get

    d^2 y
   -------  =  + .
    d^2 x


To get the value, use in calculations approximate values e = 2.71828, pi = 3.14159.


    d^2 y
   -------  =  + .
    d^2 x


With high precision online calculator WolframAlpha

https://www.wolframalpha.com/input?i=e%5Epi*cos%28e%5Epi%29-e%5E%282*pi%29*sin%28e%5Epi%29

the answer is this approximate value  479.215377365591689133.

Solved.

In this calculations, high accuracy regarding decimals is not required;
making correct formulas and showing understanding is just enough.



Answer by mccravyedwin(409)   (Show Source): You can put this solution on YOUR website!






 so 











Substitute 









 when 

 when 

Approximately 479.2153774

Edwin
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