x=tcost & y=t+sint then find dx2/d2y
To find the second derivative of x with respect to y, we need to take the derivative of x twice with respect to y. However, the given equations express x and y in terms of a parameter t, so we first need to find an expression that directly relates x and y.
From the given equations, we have x = t * cos(t) and y = t + sin(t).
To eliminate the parameter t, we can solve for t in terms of y in the second equation:
t = y - sin(t)
Substituting this value of t into the expression for x, we get:
x = (y - sin(t)) * cos(t)
Now, we can proceed to find the second derivative of x with respect to y:
Step 1: Find the first derivative of x with respect to t:
dx/dt = d/dt[(y - sin(t)) * cos(t)]
Using the product rule, we have:
dx/dt = cos(t) * d/dt(y - sin(t)) + (y - sin(t)) * d/dt(cos(t))
Simplifying this expression, we get:
dx/dt = cos(t) * (dy/dt - cos(t)) - (y - sin(t)) * sin(t)
Step 2: Find the second derivative of x with respect to t:
d^2x/dt^2 = d/dt[cos(t) * (dy/dt - cos(t)) - (y - sin(t)) * sin(t)]
Again using the product rule, we have:
d^2x/dt^2 = cos(t) * d/dt(dy/dt - cos(t)) - (dy/dt - cos(t)) * sin(t) + (y - sin(t)) * d/dt(sin(t))
Simplifying this expression, we get:
d^2x/dt^2 = cos(t) * (d^2y/dt^2 + sin(t)) - (dy/dt) * sin(t) - (dy/dt - cos(t)) * cos(t) + (y - sin(t)) * cos(t)
Step 3: Now, differentiate both sides of the equation x = (y - sin(t)) * cos(t) with respect to t to find dy/dt:
dx/dt = (dy/dt - cos(t)) * cos(t) + (y - sin(t)) * (-sin(t))
Rearranging this equation, we get:
dy/dt = (dx/dt - (y - sin(t)) * (-sin(t))) / cos(t)
Step 4: Substitute the value of dy/dt from step 3 into the expression for d^2x/dt^2 from step 2:
d^2x/d^2t = cos(t) * (d^2y/dt^2 + sin(t)) - (dy/dt) * sin(t) - (dx/dt - (y - sin(t)) * (-sin(t))) / cos(t) * cos(t) + (y - sin(t)) * cos(t)
Step 5: Simplify the expression by combining like terms and simplifying the trigonometric expressions, if necessary.
This will give you the required second derivative d^2x/d^2y.
The symbol
dx2/d2y
doesn't make sense.
Do you want d/dy (dx/dy) = d^2x/dy^2 ?
You can get dx/dy parametrically in terms of t using
dx/dy = (dx/dt)/(dy/dt)
After that, you can use the chain rule for d^2/dy^2. It will still be in terms of t.