suppose y=√(2x+1), where x and y are functions of t.
a)if dx/dt=3, find dy/dt when x=4
b)if dy/dt=5, find dx/dt when x=12
y = √(2x+1)
dy/dt = 1/√(2x+1) dx/dt
dy/dt = 1/√9 (3) = 1
5 = 1/√25 dx/dt, so dx/dt = 25
a) Well, let's find the derivative of y with respect to t. Given that y = √(2x+1), we can use the chain rule:
dy/dt = dy/dx * dx/dt
First, let's find dy/dx. Taking the derivative of y with respect to x, we get:
dy/dx = d/dx(√(2x+1))
= 1/(2√(2x+1))
Now, substituting dx/dt = 3, we have:
dy/dt = (1/(2√(2x+1))) * (dx/dt)
= (1/(2√(2(4)+1))) * 3
= (1/(2√(9))) * 3
= (1/6) * 3
= 1/2
So, when x = 4 and dx/dt = 3, dy/dt = 1/2.
b) Similarly, let's find dx/dt. We have dy/dt = 5, and we need to find dx/dt when x = 12.
Using the chain rule again, we have:
dy/dt = dy/dx * dx/dt
To find dy/dx, we differentiate y with respect to x:
dy/dx = d/dx(√(2x+1))
= 1/(2√(2x+1))
Now, substituting dy/dt = 5, we have:
5 = (1/(2√(2x+1))) * dx/dt
To find dx/dt, we rearrange the equation:
dx/dt = 5 * 2√(2x+1)
= 10√(2x+1)
Substituting x = 12, we have:
dx/dt = 10√(2(12)+1)
= 10√(25)
= 10 * 5
= 50
So, when x = 12 and dy/dt = 5, dx/dt = 50.
Hope that puts a smile on your face!
To find dy/dt when x = 4, we first need to calculate dx/dt when x = 4 using the given information.
Given: dx/dt = 3
So when x = 4, we have dx/dt = 3.
Now, let's find dy/dt using the chain rule:
dy/dt = (dy/dx) * (dx/dt)
To find dy/dx, we differentiate y with respect to x:
dy/dx = d/dx(√(2x + 1))
To differentiate a square root function, we can use the chain rule:
dy/dx = (1/2) * (2x + 1)^(-1/2) * d/dx(2x + 1)
Simplifying the expression, we have:
dy/dx = (1/2) * (2x + 1)^(-1/2) * 2
dy/dx = (2/2) * (2x + 1)^(-1/2)
dy/dx = (2/(2√(2x + 1)))
dy/dx = (1/√(2x + 1))
Now, we can substitute the given value of x = 4 into dy/dx:
dy/dx = (1/√(2(4) + 1))
dy/dx = (1/√(8 + 1))
dy/dx = (1/√9)
dy/dx = (1/3)
Substituting both dx/dt = 3 and dy/dx = 1/3 into the formula for dy/dt, we get:
dy/dt = (dy/dx) * (dx/dt)
dy/dt = (1/3) * 3
dy/dt = 1
Therefore, when x = 4, dy/dt = 1.
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To find dx/dt when x = 12, we are given that dy/dt = 5.
Now, let's find dx/dt using the chain rule:
dx/dt = (dx/dy) * (dy/dt)
To find dx/dy, we differentiate x with respect to y:
dx/dy = d/dy(√(2x + 1))
Let's differentiate using the chain rule:
dx/dy = (1/2) * (2x + 1)^(-1/2) * d/dy(2x + 1)
Simplifying the expression, we have:
dx/dy = (1/2) * (2x + 1)^(-1/2) * 2
dx/dy = (2/2) * (2x + 1)^(-1/2)
dx/dy = (2/(2√(2x + 1)))
dx/dy = (1/√(2x + 1))
Now, we can substitute the given value of y into dx/dy:
dx/dy = (1/√(2(12) + 1))
dx/dy = (1/√(24 + 1))
dx/dy = (1/√25)
dx/dy = (1/5)
Substituting both dy/dt = 5 and dx/dy = 1/5 into the formula for dx/dt, we get:
dx/dt = (dx/dy) * (dy/dt)
dx/dt = (1/5) * 5
dx/dt = 1
Therefore, when x = 12, dx/dt = 1.
To find dy/dt when x=4, we need to use the chain rule.
a) Given: dx/dt = 3, x = 4
Step 1: Find dx/dt when x = 4.
Since dx/dt = 3, we have dx/dt = 3.
Step 2: Find dy/dx.
Start with the given equation: y = √(2x + 1).
To find dy/dx, we can differentiate both sides of the equation with respect to x:
(dy/dx) = (d/dx)(√(2x + 1)).
The derivative of √(2x + 1) with respect to x can be found by applying the chain rule.
Let u = 2x + 1. Taking the derivative of u with respect to x, we have:
du/dx = 2.
Now, using the chain rule, we can find the derivative of √u with respect to x:
(dy/dx) = (d/du)(√u) * (du/dx)
(dy/dx) = (1/2√u) * 2
(dy/dx) = 1/√u
Therefore, dy/dx = 1/√(2x + 1).
Step 3: Find dy/dt.
To find dy/dt, we need to multiply dy/dx by dx/dt:
dy/dt = (dy/dx) * (dx/dt)
dy/dt = (1/√(2x + 1)) * (dx/dt)
dy/dt = (1/√(2(4) + 1)) * 3
dy/dt = (1/√9) * 3
dy/dt = (1/3) * 3
dy/dt = 1.
Therefore, when dx/dt = 3 and x = 4, dy/dt = 1.
b) Given: dy/dt = 5, x = 12
Step 1: Find dy/dt when y = 12.
Since dy/dt = 5, we have dy/dt = 5.
Step 2: Find dx/dy.
Start with the given equation: y = √(2x + 1).
To find dx/dy, we can differentiate both sides of the equation with respect to y.
Let's rewrite the given equation as x = (y^2 - 1) / 2.
Taking the derivative of x with respect to y, we get:
(dx/dy) = (d/dy)((y^2 - 1) / 2).
Using the power rule to differentiate (y^2 - 1), we have:
(dx/dy) = (1/2)(2y)
(dx/dy) = y
Therefore, dx/dy = y.
Step 3: Find dx/dt.
To find dx/dt, we need to multiply dx/dy by dy/dt:
dx/dt = (dx/dy) * (dy/dt)
dx/dt = (y) * (dy/dt)
dx/dt = 12 * 5
dx/dt = 60.
Therefore, when dy/dt = 5 and x = 12, dx/dt = 60.