Find dy/dx
x=te^t
y=t+sint
dy/dx = (dy/dt)/(dx/dt)
= (1+cost) / (e^t)(t+1)
Kind of a cool curve, with a little scaling added:
http://www.wolframalpha.com/input/?i=parametric+plot+x%3Dt%2F100+e^%28t%2F100%29%2C+y%3Dt%2F100%2Bsint
To find dy/dx, we need to differentiate y with respect to x.
Given that x = t * e^t and y = t + sin(t), we can find x in terms of t by substituting t into the equation x = t * e^t.
To differentiate y with respect to x, we need to use the chain rule since y is expressed in terms of t, and t is expressed in terms of x.
Let's start by finding dx/dt:
x = t * e^t
To differentiate x with respect to t, we can use the product rule:
dx/dt = d(t)/dt * e^t + t * d(e^t)/dt
dx/dt = 1 * e^t + t * e^t
dx/dt = e^t + t * e^t
Next, we can express dt/dx by inverting the equation x = t * e^t:
t = x / e^t
Now, let's differentiate y with respect to t:
y = t + sin(t)
dy/dt = d(t)/dt + d(sin(t))/dt
dy/dt = 1 + cos(t)
Finally, we can find dy/dx by multiplying dy/dt by dt/dx:
dy/dx = (dy/dt) * (dt/dx)
Substituting the expressions for dy/dt and dt/dx:
dy/dx = (1 + cos(t)) * (1 / (e^t + t * e^t))
Since we want to express dy/dx in terms of x, let's substitute t with x / e^t:
dy/dx = (1 + cos(x / e^t)) * (1 / (e^t + (x / e^t) * e^t))
Simplifying:
dy/dx = (1 + cos(x / e^t)) * (1 / (e^t + x))
So, the expression for dy/dx in terms of x is:
dy/dx = (1 + cos(x / e^t)) / (e^t + x)
To find dy/dx, we need to differentiate y with respect to x. In this case, x and y are both functions of t, so we can use the chain rule to differentiate y with respect to t and then multiply by dt/dx.
Step 1: Differentiate y with respect to t (dy/dt):
Since y = t + sin(t), the derivative of y with respect to t is dy/dt = 1 + cos(t).
Step 2: Differentiate x with respect to t (dx/dt):
Since x = t * e^t, we have two terms: the derivative of t with respect to t is 1, and the derivative of e^t with respect to t is e^t. Therefore, dx/dt = e^t + t * e^t.
Step 3: Multiply dy/dt by dt/dx:
To find dy/dx, we multiply dy/dt by dt/dx. Since dt/dx is the reciprocal of dx/dt, we have dt/dx = 1 / (e^t + t * e^t).
dy/dx = (dy/dt) * (dt/dx)
= (1 + cos(t)) * (1 / (e^t + t * e^t))
Thus, dy/dx = (1 + cos(t)) / (e^t + t * e^t).