If x = cost + logtant/2 and y = sint, then find the values of d^2y/dt^2 and d^2y/dx^2 at t - Maths - Continuity and Differentiability

Question

If x = cost + logtant/2 and y = sint, then find the values of
d^2y/dt^2 and d^2y/dx^2 at t = pie/4

Mayur Pisode · Accepted Answer

We have,      x = cos t + log tan t2⇒dxdt = - sin t  + 12× cott/2 
 sec2t/2⇒dxdt = - sin t + 12 sin t/2 
 cos t/2⇒dxdt = -sin t + 1sin t⇒dxdt = 1 - sin2tsin t⇒dxdt = cos2tsin tNow,        y = sin t⇒dydt = cos tNow, dydx = dydt ×dtdx⇒ dydx = cos t × sin tcos2t⇒dydx = tan t⇒d2ydx2 = sec2t × dtdx ⇒d2ydx2 =  sec2t × sin tcos2t⇒d2ydx2 = sin tcos4t⇒d2ydx2t=π/4 = sinπ/4cos4π/4 = 1214 = 12 × 41 = 22Now, dydt = cos t⇒d2ydt2 = - sin t⇒d2ydt2t=π/4 = - sinπ/4 = -12

If x = cost + logtant/2 and y = sint, then find the values of d^2y/dt^2 and d^2y/dx^2 at t = pie/4