Find the second derivative (d2y/dx2) for the curve given by
x = 2·cos() and y = sin()
To find the second derivative (d2y/dx2) for the curve, we need to differentiate the equation of y with respect to x twice.
Let's start by finding the first derivative (dy/dx) using the chain rule of differentiation.
Given:
x = 2·cos()
y = sin()
To find dy/dx, we differentiate y with respect to the variable inside the sine function and multiply by the derivative of the variable with respect to x:
dy/dx = (dy/d() ) · (d()/dx)
Since y = sin(), the derivative of sine is cosine:
dy/dx = cos() · (d()/dx)
Now, to find d()/dx, we differentiate x = 2·cos() with respect to x:
d()/dx = d(2·cos() )/dx
The derivative of cosine with respect to x is:
d(cos() )/dx = -sin()
Substituting this value back into the equation for dy/dx:
dy/dx = cos() · -sin() = -sin() · cos()
Now, to find the second derivative (d2y/dx2), we differentiate dy/dx with respect to x:
d2y/dx2 = d(dy/dx)/dx
To find d(dy/dx)/dx, we differentiate -sin() · cos() with respect to x:
d(-sin() · cos() )/dx
We differentiate each term separately:
d(-sin() )/dx = -cos()
d(cos() )/dx = -sin()
Now, we can substitute these values back into our equation:
d(-sin() · cos() )/dx = -cos() · (-sin()) - sin() · (-cos())
= sin() · cos() - sin() · cos()
= 0
Therefore, the second derivative (d2y/dx2) of the given curve is 0.
To find the second derivative (d^2y/dx^2) for the curve given by x = 2·cosθ and y = sinθ, we need to differentiate the equation of y twice with respect to x.
Step 1: Find the first derivative of y with respect to x.
We can first find dy/dθ, then multiply it by dθ/dx using the chain rule.
dy/dθ = cosθ (derivative of sinθ with respect to θ)
dθ/dx = 1 / (dx/dθ) = 1 / (-2sinθ) (derivative of x = 2cosθ with respect to θ)
dy/dx = (dy/dθ) * (dθ/dx) = cosθ * (1 / (-2sinθ)) = -1/2
Step 2: Find the second derivative of y with respect to x.
This involves differentiating dy/dx with respect to x.
d^2y/dx^2 = d(dy/dx)/dx
= d(-1/2)/dx
= 0
Therefore, the second derivative (d^2y/dx^2) for the curve x = 2·cosθ and y = sinθ is equal to 0.