The demand x is the number of items that can be sold at a price of $p. For
x = p^3 - 4p + 400,
find the rate of change of p with respect to x by differentiating implicitly.
Answer
To find the rate of change of p with respect to x, we need to differentiate the equation implicitly.
Taking the derivative of both sides of the equation with respect to x:
d/dx(x) = d/dx(p^3 - 4p + 400)
Simplifying, we get:
1 = (3p^2 - 4) * dp/dx
Now, we can isolate dp/dx:
dp/dx = 1 / (3p^2 - 4)
So the rate of change of p with respect to x is given by dp/dx = 1 / (3p^2 - 4).
To find the rate of change of p with respect to x by differentiating implicitly, we can apply the implicit differentiation technique. Here's how you can do it step by step:
Step 1: Start with the given equation: x = p^3 - 4p + 400.
Step 2: Differentiate both sides of the equation with respect to x. Remember that we are treating p as a function of x, so we need to use the chain rule when differentiating terms with p.
d/dx(x) = d/dx(p^3 - 4p + 400)
Step 3: Differentiate each term with respect to x.
The derivative of x with respect to x is simply 1:
1 = d/dx(p^3) - d/dx(4p) + d/dx(400)
Step 4: Apply the power rule to differentiate p^3.
To differentiate p^3 with respect to x, we apply the power rule, which states that the derivative of x^n is n * x^(n-1):
1 = 3p^2 * d/dx(p) - 4 * d/dx(p) + 0
Step 5: Simplify the equation:
1 = 3p^2 * dp/dx - 4 * dp/dx
Step 6: Factor out the dp/dx term:
1 = dp/dx * (3p^2 - 4)
Step 7: Solve for dp/dx:
dp/dx = 1 / (3p^2 - 4)
So, the rate of change of p with respect to x is given by dp/dx = 1 / (3p^2 - 4).
1 = 3p^2 dp/dx - 4 dp/dx
1 = (3p^2-4) dp/dx
dp/dx = 1/(3p^2-4)
also, note that
dx/dp = 3p^2-4
and dp/dx = 1/(dx/dp)