suppose a price-demand function p(x) is defined implicitly by
x= p^3-3p^2+200
find p'.
p'=1/x'
x'=3p^2-6p
To find the derivative of the function p(x), we can use the implicit differentiation method. Let's start by differentiating both sides of the equation with respect to x.
Hold on! Before we proceed, let's rename the variables to avoid confusion. Instead of using p for both the function and the variable, let's use y for the function and p for the variable. So, we have:
x = p^3 - 3p^2 + 200
Now, we can differentiate both sides of the equation with respect to x. Keep in mind that when we differentiate p^3, we need to use the chain rule.
Differentiating both sides:
d/dx(x) = d/dx(p^3 - 3p^2 + 200)
Since d/dx(x) is simply 1, we have:
1 = d/dx(p^3) - d/dx(3p^2) + d/dx(200)
Applying the chain rule to d/dx(p^3) and the power rule to d/dx(3p^2):
1 = 3p^2 * dp/dx - 6p * dp/dx + 0
Simplifying the expression:
1 = 3p^2 * dp/dx - 6p * dp/dx
Now let's rearrange the equation to solve for dp/dx, which is the derivative of p(x) with respect to x:
1 = (3p^2 - 6p) * dp/dx
Dividing both sides by (3p^2 - 6p):
dp/dx = 1 / (3p^2 - 6p)
Now we have the derivative of p(x), expressed in terms of p.