A point is moving along the graph of y = 1/(1 + x^2) so that dxldt = 2 centimeters per minute. Find dy/dt for the following values of x.
(a) X = -2
well,
dy/dt = -2x/(1+x^2)^2 dx/dt
so plug in x = -2
To find dy/dt when x = -2, we need to differentiate the equation y = 1/(1 + x^2) with respect to t using the chain rule.
First, let's differentiate y = 1/(1 + x^2) with respect to x:
dy/dx = d/dx (1/(1 + x^2))
= -1/(1 + x^2)^2 * d/dx (1 + x^2)
= -1/(1 + x^2)^2 * (2x)
= -2x/(1 + x^2)^2
Next, we'll use the chain rule to calculate dy/dt:
dy/dt = dy/dx * dx/dt
Given dx/dt = 2 (centimeters per minute), we have:
dy/dt = -2x/(1 + x^2)^2 * dx/dt
= -2(-2)/(1 + (-2)^2)^2 * 2
= -4/9
Therefore, when x = -2, dy/dt = -4/9.
To find dy/dt for a specific value of x, we'll need to differentiate the function y = 1/(1 + x^2) with respect to t and then plug in the given value of x.
Step 1: Differentiate y with respect to t using the chain rule.
To differentiate y with respect to t, we need to apply the chain rule. Recall that the chain rule states that if y = f(u) and u = g(x), then dy/dx = dy/du * du/dx.
In our case, y = f(u), where f(u) = 1/u, and u = g(x), where g(x) = 1 + x^2. So we have:
dy/dt = dy/du * du/dx * dx/dt
Step 2: Find dy/du.
Differentiating f(u) = 1/u with respect to u gives us:
df/du = -1/u^2
Step 3: Find du/dx.
Differentiating g(x) = 1 + x^2 with respect to x gives us:
dg/dx = 2x
Step 4: Find dx/dt.
We are given that dx/dt = 2 cm/min.
Step 5: Plug in the values and solve.
Now, we have all the necessary values to calculate dy/dt.
dy/dt = dy/du * du/dx * dx/dt
= (-1/u^2) * (2x) * (2 cm/min)
For x = -2, we can substitute it into this equation and calculate dy/dt:
dy/dt = (-1/(1 + x^2)^2) * (2x) * (2 cm/min)
= (-1/(1 + (-2)^2)^2) * (2(-2)) * (2 cm/min)
= (-1/(1 + 4)^2) * (-4) * (2 cm/min)
= (-1/25) * (-4) * (2 cm/min)
= 8/25 * 4 cm/min
= 32/25 cm/min
Therefore, for x = -2, dy/dt is equal to 32/25 cm/min.