suppose u and v are functions of x that are differentiable at x=2 and
that u(2) =3, u'(2) = -4, v(2) = 1, and v'(2)
find values of derivatives at x = 2
(d/dx)(uv) = ?
I would like to know how to set this up because I'm only used to getting problems that want the d/dx given ex: y=2x+1 so I was confused for this
The answer is 2 but how do I set this up?
To find the value of the derivative at x = 2 for the function (d/dx)(uv), you can use the product rule for differentiation.
The product rule states that the derivative of the product of two functions u(x) and v(x) is given by:
(d/dx)(uv) = u'v + uv'
Now, let's plug in the given information to find the values of u'(2) and v'(2):
u'(2) = -4 (given)
v'(2) = ... this value is missing in the question.
Since the value of v'(2) is missing, we don't have complete information to calculate the derivative at x = 2 for (d/dx)(uv). We need to know the value of v'(2) to proceed further.
Once you have the value of v'(2), you can calculate the derivative by substituting the given values into the product rule formula:
(d/dx)(uv) = u'(2) * v(2) + u(2) * v'(2)
Given that u(2) = 3, u'(2) = -4, v(2) = 1, and v'(2) = ? (which you need to provide), you can substitute in these values to find the derivative at x = 2:
(d/dx)(uv) = (-4) * 1 + 3 * v'(2)
Simplifying further:
(d/dx)(uv) = -4 + 3 * v'(2)
Without knowing the value of v'(2), we cannot simplify this expression any further or evaluate it at x = 2. Therefore, the answer to the question cannot be determined at this point without the missing information.
To find the value of the derivative of the product of u(x) and v(x) at x = 2, you can use the product rule of differentiation. The product rule states that if y = u(x) * v(x), then the derivative of y with respect to x, denoted as dy/dx, is given by:
dy/dx = u'(x) * v(x) + u(x) * v'(x)
In this case, you are given u(2) = 3, u'(2) = -4, v(2) = 1, and v'(2) is unknown. To find the value of the derivative at x = 2, substitute these values into the product rule:
(d/dx)(uv) = u'(2) * v(2) + u(2) * v'(2)
Now, substitute the given values:
(d/dx)(uv) = (-4)(1) + (3)(v'(2))
Simplify the expression:
(d/dx)(uv) = -4 + 3v'(2)
Since the value of the derivative at x = 2 is given as 2, you can set -4 + 3v'(2) equal to 2:
-4 + 3v'(2) = 2
Solve this equation for v'(2):
3v'(2) = 6
Divide both sides by 3:
v'(2) = 2
Therefore, the value of the derivative of uv at x = 2 is 2.
This is just the product rule.
d/dx(uv) = du/dx * v + u * dv/dx
when x=2,
d/dx(uv) = (-4)(1) + (3)(2) = -4+6 = 2