Find
dy/dt
at x = 1 and y = x2 + 3 if
dx/dt
= 4
I'm not sure how to set up this type of equation. Do you use the pythagorem theorem for this? Then take the derivative, and then plug in?
y = x^2+3
dy/dt = 2x dx/dt = 2*1*4 = 8
Oh dear, Pythagorean theorem won't be much help here. But, fear not! I'm here to guide you through this math maze with a smile.
We're given that dx/dt = 4. However, we need to find dy/dt at x = 1 and y = x^2 + 3.
To solve this, we'll start by finding dy/dt. Since y = x^2 + 3, let's differentiate both sides with respect to t:
dy/dt = d/dt (x^2 + 3)
Now, since dx/dt = 4, we can rewrite the derivative as:
dy/dt = d/dt (x^2 + 3) = d/dx (x^2 + 3) * dx/dt
The derivative of x^2 + 3 with respect to x is 2x. So, plugging in dx/dt = 4 and x = 1, we get:
dy/dt = 2x * dx/dt = 2(1) * 4 = 8
Voila! Thus, at x = 1 and y = x^2 + 3, dy/dt is equal to 8. No Pythagorean theorem necessary, just a little derivative fun.
To find dy/dt at the given values, we need to use the chain rule to differentiate the equation y = x^2 + 3 with respect to t.
The chain rule states that if we have y = f(u) and u = g(t), then dy/dt = dy/du * du/dt.
In this case, u = x and f(u) = u^2 + 3. So, we can rewrite the equation as y = u^2 + 3 and find dy/du.
Differentiating y = u^2 + 3 with respect to u gives us dy/du = 2u.
Now, we need to find du/dt, which is given as dx/dt = 4.
Finally, using the chain rule, we can find dy/dt as follows:
dy/dt = dy/du * du/dt
= 2u * dx/dt
= 2(1) * 4
= 8.
Therefore, dy/dt at x = 1 and y = x^2 + 3, with dx/dt = 4, is 8.
To find dy/dt at x = 1 and y = x^2 + 3, we can use the chain rule of differentiation. The Pythagorean theorem is not applicable in this case.
First, we are given dx/dt = 4, which represents the rate of change of x with respect to t. This means that for every unit of t that passes, x changes by 4 units.
We want to find dy/dt, the rate of change of y with respect to t. To do this, we can use the chain rule, which states that dy/dt = dy/dx * dx/dt.
Let's start by finding dy/dx. We can take the derivative of y = x^2 + 3 with respect to x:
dy/dx = 2x
Now, we have dy/dx and dx/dt, so we can substitute these values into the chain rule equation:
dy/dt = (dy/dx) * (dx/dt)
= (2x) * (4)
From this, we can see that dy/dt = 8x.
Now, we need to find dy/dt at x = 1. We substitute x = 1 into the equation:
dy/dt = 8(1)
= 8
Therefore, dy/dt at x = 1 and y = x^2 + 3, when dx/dt = 4, is 8.