1/f = 1/p + 1/q
What is the rate of change of p with respect to q if q=2 and f=4?
nvm i figured it out
To find the rate of change of p with respect to q, we'll need to differentiate the equation 1/f = 1/p + 1/q with respect to q. This will allow us to determine how p changes as q changes.
First, rewrite the equation as follows:
1/f = 1/p + q/(pq)
To differentiate the equation, we'll apply the quotient rule, which states that if we have a function of the form f(x) = g(x)/h(x), then the derivative of f(x) with respect to x is given by:
f'(x) = (g'(x) * h(x) - g(x) * h'(x)) / (h(x))^2
Now let's differentiate both sides of the equation with respect to q and solve for dp/dq:
d/dq (1/f) = d/dq (1/p + q/(pq))
To differentiate 1/f, we need to apply the chain rule because f is a function of q. The derivative of 1/f with respect to q is:
-(1/f^2) * df/dq
Since f = 4, df/dq = 0 because f is a constant with respect to q when f = 4.
Now differentiating 1/p + q/(pq) with respect to q, we get:
d/dq (1/p + q/(pq)) = 0 + (1/p^2) - (q/(p^2 * q^2)) = 1/p^2 - 1/(p^2q^2)
Now we can substitute the respective values for p, q, and f:
dp/dq = (1/p^2 - 1/(p^2q^2)) * (1/f^2) = (1/p^2 - 1/(p^2q^2)) * (1/4^2) = (1/p^2 - 1/(p^2q^2)) * (1/16)
Since q = 2 and f = 4, we can simplify the equation further:
dp/dq = (1/p^2 - 1/(p^2 * 2^2)) * (1/16) = (1/p^2 - 1/(4p^2)) * (1/16)
Now, substitute the value of q = 2 and f = 4 into the equation:
dp/dq = (1/p^2 - 1/(4p^2)) * (1/16) = (1/p^2 - 1/(4p^2)) * (1/16) = (1/p^2 - 1/(4p^2)) * (1/16)
= (4 - 1)/(4p^2) * (1/16) = (3)/(4p^2) * (1/16) = (3)/(64p^2)
So, the rate of change of p with respect to q, when q = 2 and f = 4, is 3/(64p^2).