Question 1208213
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if y = sin (t) , x = ln (t) , then ,(d ^2 y)/(dx ^2)=... when (x = π)
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<pre>
If  x = ln(t),  then  

    t = {{{e^x}}},      (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

    {{{(dy)/dx)}}} = {{{cos(e^x)*((d(e^x))/(dx))}}} = {{{cos(e^x)*e^x}}}.    (3)


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

    d^2 y
   ------- (x) = {{{-sin(e^x)*e^x*e^x}}} + {{{cos(e^x)*e^x}}} = {{{-sin(e^x)*e^(2x)}}} + {{{cos(e^x)*e^x}}}.
    d^2 x


Now we substitute x = {{{pi}}} to get

    d^2 y
   ------- {{{(pi)}}} = {{{-sin(e^pi)*e^(2pi)}}} + {{{cos(e^pi)*e^pi}}}.
    d^2 x


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


    d^2 y
   ------- {{{(pi)}}} = {{{-sin(2.71828^3.14159)*2.71828^(2*3.14159)}}} + {{{cos(2.71828^3.14159)*2.71828^3.14159}}}.
    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.
</pre>

Solved.


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