If x = t^2 + 1 and y = t^3, then d^2y/dx^2 =
I know I can solve for t in terms of x and substitute that into y = t^3 and find the double derivative.
I also know that I can take the derivative of x and y then divide dy/dt by dx/dt. Then take the derivative to get the answer.
However, my question is why can't you find d^2x/dt^2 and d^2y/dt^2 and divide to find d^2y/dx^2?
Like
x = t^2 + 1
dx/dt = 2t(dt/dt) + 0(dt/dt)
d^2x/dt^2 = 2(dt/dt) = 2
y = t^3
dy/dt = 3t^2(dt/dt)
d^2y/dt^2 = 6t(dt/dt)
d^2y/dx^2 = (d^2y/dt^2)/(d^2x/dt^2)
d^2y/dx^2 = 6t / 2 = 3t
The correct answer, however, is 3/4t. Why? Am I doing something wrong -.-?
See this
http://en.wikipedia.org/wiki/Parametric_derivative
Ah, I see. Just for confirmation, d^2x != dx^2, correct?
You made a small mistake in your calculations. When you take the second derivative of x with respect to t, you correctly found that d^2x/dt^2 = 2. However, when you take the second derivative of y with respect to t, you need to consider the chain rule.
Let's go through the calculations step by step:
Given:
x = t^2 + 1
y = t^3
Step 1: Find dx/dt:
dx/dt = 2t
Step 2: Find dy/dt:
dy/dt = 3t^2
Step 3: Find d^2x/dt^2:
d^2x/dt^2 = d/dt (dx/dt) = 2
Step 4: Find d^2y/dt^2:
d^2y/dt^2 = d/dt (dy/dt) = d/dt (3t^2) = 6t
Step 5: Find d^2y/dx^2:
We need to find dy/dx first.
dy/dx = (dy/dt) / (dx/dt) = (3t^2) / (2t) = 3t/2
Now, let's find the second derivative:
d^2y/dx^2 = d/dx (dy/dx) = d/dx (3t/2)
To find d/dx, we need to express t in terms of x:
x = t^2 + 1
t^2 = x - 1
t = sqrt(x - 1)
Now, let's substitute this back into the expression:
d^2y/dx^2 = d/dx (3(sqrt(x - 1))/2)
= 3/2 * (1/2) * (1/sqrt(x - 1))
= 3/4 * (1/sqrt(x - 1))
= 3/(4sqrt(x - 1))
So, the correct answer is d^2y/dx^2 = 3/(4sqrt(x - 1)).
To find d^2y/dx^2, we need to use the chain rule. The chain rule states that if you have a function y = f(g(x)), then the derivative of y with respect to x is given by dy/dx = f'(g(x)) * g'(x).
In this case, we have x = t^2 + 1 and y = t^3. To find dy/dt, we can simply take the derivative of y with respect to t:
dy/dt = d/dt(t^3) = 3t^2
Now, let's find dx/dt. Taking the derivative of x = t^2 + 1 with respect to t:
dx/dt = d/dt(t^2 + 1) = 2t
To find d^2y/dx^2, we need to find the derivative dy/dx. To do this, we will use the chain rule by dividing dy/dt by dx/dt:
dy/dx = (dy/dt) / (dx/dt) = (3t^2) / (2t) = 3t/2
Now, let's find d^2y/dx^2. To do this, we need to take the derivative of dy/dx with respect to x. Using the chain rule again, we get:
d^2y/dx^2 = (d/dx)(dy/dx) = (d/dt)(dy/dx) / (dx/dt)
= (d/dt)(3t/2) / (2t)
= (3/2) / 2
= 3/4
So the correct answer is indeed 3/4, not 3t. It seems there was a mistake in your computation.