Assume that x>0 and y>0. If y=5sqrt{x+9} and dx/dt = 2, find dy/dt when x=7.

y= 5sqrt(x+9)

dy/dt=1/2 *5 *1/sqrt(x+9) * dx/dt
put the values in, and find dy/dt

To find dy/dt when x=7, we need to find the derivative of y with respect to t and then substitute x=7.

First, let's differentiate y with respect to t using the chain rule. Recall that the chain rule states that if y = f(u) and u = g(x), then dy/dx = f'(u) * g'(x).

In this case, let u = x + 9 and f(u) = 5sqrt(u). Therefore, y = f(u) = 5sqrt{x + 9}.

To find dy/dt, we need to find dy/dx * dx/dt. We already know dx/dt = 2.

Now, let's find dy/dx. We can do this by applying the power rule and the chain rule. The derivative of sqrt(u) with respect to u is (1/2)sqrt(u), and the derivative of 5u^1/2 is (5/2)sqrt(u).

Therefore, dy/dx = (5/2)sqrt(x + 9).

Now, we can find dy/dt by substituting x=7 and dx/dt=2 into the equation dy/dt = dy/dx * dx/dt.

dy/dt = (5/2)sqrt(7 + 9) * 2 = (5/2)sqrt(16) * 2 = (5/2)*4*2 = 20

So, dy/dt when x=7 is 20.