Question

x=cost^t and y=sint^t
How to solve this differentiation?

Answer 1

Hello Friend,

Here,

x = cos t^t
y = sin t^t

This is an example of parametric function, and here, t is the parameter.

To find dy/dx, we can individually find dx/dt and dy/dt. To differentiate, we will use chain rule.

→ x = cos t^t

• Let t^t = u
So, log t^t = log u
So, t log t = log u

Differentiating with respect to t

t . (1/t) + (1) log t = (1/u) du/dt
So, du/dt = u(1 + log t)
So, du/dt = t^t (1 + log t)

• Now, x = cos t^t
So, x = cos u
So, dx/du = - sin u

• dx/dt = (dx/du) (du/dt)
So, dx/dt = (-sin u) [t^t(1+log t)]

So, dx/dt = - t^t(1 + log t) sin t^t

→ Similarly,

y = sin t^t
So, y = sin u
So, dy/du = cos u

• Now, dy/dt = (dy/du) (du/dt)
So, dy/dt = cos u [t^t (1 + log t)]
So, dy/dt = t^t (1 + log t) cos t^t

→ Now,

dy/dx
= (dy/dt) ÷ (dx/dt)
= [ t^t (1 + log t) cos t^t ] ÷ [- t^t(1 + log t) sin t^t]
= - cot t^t

→ Thus, dy/dx = - cot t^t


Hope it helps.

Purva
@Purvaparmar1405
Brainly.in