Question

If x = a (cos 2t + 2t sin 2t) and y = a (sin 2t – 2t cos 2t), then find 2 2 dx d y .

Answer 1

$\frac{d^2y}{dx^2}=\frac{1}{2at}sec^32t$

Step-by-step explanation:

$x=a(cos2t+2tsin2t)$

$y=a(sin2t-2tcos2t)$

Differentiate w.r.t t

$\frac{dx}{dt}=a(-2sin2t+2sin2t+4tcos2t)$

By using the formula

$\frac{d(cosx)}{dx}=-sinx$

$\frac{d(sinx)}{dx}=cosx$

$\frac{d(u\cdot v)}{dx}=u'v+v'u$

$\frac{dx}{dt}=4atcos2t$

$\frac{dy}{dt}=a(2cos2t-2cos2t+4tsin2t)=4atsin2t$

$\frac{dy}{dx}=\frac{\frac{dy}{dt}}{\frac{dx}{dt}}$

$\frac{dy}{dx}=\frac{4atsin2t}{4atcos2t}=tan2t$

By using $\frac{sinx}{cosx}=tanx$

$\frac{d^2y}{dx^2}=\frac{d(\frac{dy}{dx})}{dx}=\frac{d(tan2t)}{dx}=sec^22t\times 2\times \frac{dt}{dx}=2sec^22t\times \frac{1}{4atcos2t}$

By using the formula $\frac{d(tanx)}{dx}=sec^2x$

$\frac{d^2y}{dx^2}=\frac{1}{2at}sec^22t\times sec2t=\frac{1}{2at}sec^32t$

By using $\frac{1}{cosx}=secx$

#Learns more:

https://brainly.com/question/2254613

Answer 2

Answer:

Step-by-step explanation: