Suppose
y = sqrt(2x + 1), where x and y are functions of t.
(a) If dx/dt = 15, find dy/dt when x = 4.
dy/dt = ?
dy/dt = 1/(2√(2x+1)) * 2 * dx/dt
so, at the given point,
dy/dt = 1/(2√9) * 2 * 15 = 5
To find dy/dt, we need to differentiate y with respect to t using the chain rule.
Given: y = sqrt(2x + 1)
Step 1: Find dx/dt
Given: dx/dt = 15
Step 2: Substitute x = 4 into dx/dt
dx/dt = 15
Step 3: Differentiate y with respect to t using the chain rule
(dy/dt) = (dy/dx) * (dx/dt)
Step 4: Find dy/dx.
To find dy/dx, differentiate y = sqrt(2x + 1) with respect to x.
(dy/dx) = (1/2) * (2x + 1)^(-1/2) * (2)
Step 5: Substitute dy/dx and dx/dt into the equation dy/dt = (dy/dx) * (dx/dt)
(dy/dt) = (1/2) * (2x + 1)^(-1/2) * (2) * (dx/dt)
Step 6: Substitute x = 4 and dx/dt = 15
(dy/dt) = (1/2) * (2(4) + 1)^(-1/2) * (2) * 15
Step 7: Simplify the equation
(dy/dt) = 15 * (2(4) + 1)^(-1/2)
Step 8: Evaluate the expression
(dy/dt) = 15 * (9)^(-1/2)
Step 9: Simplify the square root
(dy/dt) = 15 * (1/3)
Step 10: Calculate the final answer
(dy/dt) = 15/3
Therefore, dy/dt = 5.
To find dy/dt, we can use the chain rule, which states that for a composite function y = f(g(x)), the derivative of y with respect to x is given by dy/dx = f'(g(x)) * g'(x).
In this case, we have y = sqrt(2x + 1), where x and y are functions of t. To find dy/dt, we need to find dy/dx and dx/dt, and then multiply them together.
(a) Given dx/dt = 15, we need to find dy/dt when x = 4.
Step 1: Find dx/dt
Given dx/dt = 15, we already know the value. It is 15.
Step 2: Find dy/dx
To find dy/dx, we differentiate y with respect to x.
Let's start by rewriting y = sqrt(2x + 1) as y = (2x + 1)^(1/2).
Using the power rule, we can differentiate this function with respect to x.
dy/dx = (1/2) * (2x + 1)^(-1/2) * 2
= (2/(2x + 1))^(1/2)
So dy/dx = (2/(2x + 1))^(1/2).
Step 3: Calculate dy/dt
Now, we can calculate dy/dt by multiplying dy/dx and dx/dt.
dy/dt = (2/(2x + 1))^(1/2) * dx/dt
In this case, x = 4:
dy/dt = (2/(2(4) + 1))^(1/2) * 15
= (2/9)^(1/2) * 15
= (2^(1/2) /9^(1/2)) * 15
= (2/3) * 15
= 10
Therefore, when x = 4 and dx/dt = 15, dy/dt = 10.