Find
dy/dx and d2y/dx2.
x = e^t, y = te^t
x=e^t
dx/dt=e^t
y=te^t
dy/dt=e^t+te^t
dy/dx=dy/dt / dx/dt=(e^t)/(e^t(1+t))= 1/(1+t)
y"/dx"=D (1/(1+t)=-1/(1+t)^2
not so. you have found d/dt (dy/dx), but you want
d/dx (dy/dx) = d/dt(dy/dx) / dx/dt
= -1/(1+t)^2 / e^t = -1/((1+t)^2 e^t)
To find dy/dx, we need to use the chain rule. The chain rule states that if y is a function of u and u is a function of x, then dy/dx = dy/du * du/dx.
In this case, x = e^t and y = te^t. We need to find dy/dx.
Step 1: Find dy/du
To find dy/du, we differentiate y with respect to u. Since y = te^t, we can rewrite it as y = t * (e^t) and treat t as a constant when differentiating with respect to u.
dy/du = t * d(e^t)/du
To differentiate e^t with respect to u, we need to use the chain rule again, which states that if u is a function of t and t is a function of u, then d/dt * du/dx = du/dt.
Since u = e^t, du/dt = d(e^t)/dt = e^t.
Using the chain rule, d(e^t)/du = (d(e^t)/dt) * (dt/du) = e^t.
Therefore, dy/du = t * e^t.
Step 2: Find du/dx
To find du/dx, we differentiate x = e^t with respect to t.
du/dx = d(e^t)/dt
Differentiating e^t with respect to t gives us du/dx = e^t.
Step 3: Find dy/dx
Now that we have dy/du = t * e^t and du/dx = e^t, we can find dy/dx by multiplying these two derivatives.
dy/dx = (dy/du) * (du/dx)
= (t * e^t) * (e^t)
= t * e^t * e^t
= t * e^2t.
Therefore, dy/dx = t * e^2t.
To find d^2y/dx^2, we need to find the second derivative of y with respect to x.
Step 1: Find the first derivative of dy/dx
We already found dy/dx to be t * e^2t.
Step 2: Differentiate dy/dx with respect to x to get the second derivative.
d^2y/dx^2 = d(dy/dx)/dx
Differentiating t * e^2t with respect to x gives us d^2y/dx^2 = (d(t * e^2t)/dt) * (dt/dx).
Since we know that x = e^t, we can rewrite dt/dx as 1/(dx/dt).
Step 3: Find dt/dx
To find dt/dx, we need to take the derivative of x = e^t with respect to t.
dx/dt = d(e^t)/dt
Differentiating e^t with respect to t gives us dx/dt = e^t.
Step 4: Find d^2y/dx^2
Now that we have d(t * e^2t)/dt = e^2t and dt/dx = e^t, we can find d^2y/dx^2 by multiplying these two derivatives.
d^2y/dx^2 = (d(t * e^2t)/dt) * (dt/dx)
= e^2t * e^t
= e^(2t+t)
= e^(3t).
Therefore, d^2y/dx^2 = e^(3t).