q=(1000/p+1)^2 - 1
find q'
Please help. Question on review sheet.
I will be glad to help you, or critique your work. What do you think the way to solve it is>
To find the derivative of q, we can use the power rule for differentiation. The power rule states that if we have a function of the form f(x) = (g(x))^n, then the derivative of f(x) is given by f'(x) = n(g(x))^(n-1) * g'(x).
In our case, q is given as q = (1000/p + 1)^2 - 1. To find q', we need to differentiate each term separately.
First, let's differentiate the first term: (1000/p + 1)^2. Using the chain rule, we need to differentiate the outer function (g(x) = x^2) and then multiply by the derivative of the inner function (h(p) = 1000/p + 1). The derivative of g(x) = x^2 is g'(x) = 2x, and the derivative of h(p) = 1000/p + 1 can be found using the quotient rule or the power rule for differentiation (if we rewrite h(p) as h(p) = 1000/p + p^(-1)).
Differentiating h(p) = 1000/p + p^(-1) using the quotient rule, we get:
h'(p) = (p * (d/dp)(1000/p) - (1000/p + p^(-1)) * (d/dp)(p)) / (p^2)
Simplifying this expression, we have:
h'(p) = (-1000/p^2 - 1) / (p^2)
Multiplying the derivative of the outer function by the derivative of the inner function, we get:
(g'(h(p)) * h'(p)) = 2 * ((1000/p + 1)^2) * (-1000/p^2 - 1) / (p^2)
Next, let's differentiate the second term: -1. The derivative of a constant is always zero.
Now, we can add the derivatives of each term to find q':
q' = 2 * ((1000/p + 1)^2) * (-1000/p^2 - 1) / (p^2) + 0
Simplifying this expression will give us the final derivative of q.