Second derivative of y relative to x^2

Question

^{y = cos(t)}

t = sin(x)

 

for x =π what's the value of d²y over dx²

Answer 1

y = cos ( sin(x))

 

dy/dx =   sin ( sin(x)) cos(x)  <--- chain rule

 

 

Product Rule/Chain Rule needed for 2nd derivative.

 

   sin (sin(x))* -sin(x) +  cos(sin(x))*cos(x)*cos(x)

 

   sin( sin(x))*sin(x) + cos(sin(x)) * cos(x)^2

 

 

For x = pi, the first term in bold of the second derivative vanishes

 because sin(pi) = 0.

 

 So the second derivative at x=pi is cos(0) * cos(pi)^2

                                                     1*(-1)^2 = 1