Find the first partial derivatives and evaluate each at the given point.
Function
w = sqrt(5x^2 + y^2 − 4z^2)
Point
(2, −4, 2)
wx(2, −4, 2) =
wy(2, −4, 2) =
wz(2, −4, 2) =
w = (5 x^2 + y^2 - 4 z^2)^.5
dw/dx = (5 x)/(5 x^2 + y^2 - 4 z^2)^.5
dw/dy = y/(5 x^2 + y^2 - 4 z^2)^.5
dw/dz = -4/(5 x^2 + y^2 - 4 z^2)^.5
now just plug in 2, -4, 6
make that -4z...
But I'm sure the student caught the typo.
Thanks, hope so :)
To find the partial derivatives of the given function and evaluate them at the given point (2, -4, 2), we need to differentiate the function with respect to each variable and then substitute the values of x, y, and z with the given point.
1. wx denotes the partial derivative of w with respect to x.
To find wx, we differentiate the function sqrt(5x^2 + y^2 - 4z^2) with respect to x:
wx = (1/2)(5x^2 + y^2 - 4z^2)^(-1/2) * d/dx (5x^2 + y^2 - 4z^2)
Now, let's differentiate the expression inside the parentheses with respect to x:
d/dx (5x^2 + y^2 - 4z^2) = 10x.
Substituting the given point (2, -4, 2) into wx:
wx(2, -4, 2) = (1/2)(5(2)^2 + (-4)^2 - 4(2)^2)^(-1/2) * 10(2)
= (1/2)(20 + 16 - 16)^(-1/2) * 20
= (1/2)(20)^(-1/2) * 20
= (1/2)(1/√20) * 20
= 1/√20 * 10
= 10/√20
= (10/√20) * (√20/√20)
= 10√20/20
= √20/2
= (√4*√5)/2
= 2√5/2
= √5
Therefore, wx(2, -4, 2) = √5.
2. wy denotes the partial derivative of w with respect to y.
To find wy, we differentiate the function sqrt(5x^2 + y^2 - 4z^2) with respect to y:
wy = (1/2)(5x^2 + y^2 - 4z^2)^(-1/2) * d/dy (5x^2 + y^2 - 4z^2)
Now, let's differentiate the expression inside the parentheses with respect to y:
d/dy (5x^2 + y^2 - 4z^2) = 2y.
Substituting the given point (2, -4, 2) into wy:
wy(2, -4, 2) = (1/2)(5(2)^2 + (-4)^2 - 4(2)^2)^(-1/2) * 2(-4)
= (1/2)(20 + 16 - 16)^(-1/2) * (-8)
= (1/2)(20)^(-1/2) * (-8)
= (1/2)(1/√20) * (-8)
= -8/√20
= (-8/√20) * (√20/√20)
= -8√20/20
= -√20/2
= -√(4*5)/2
= -2√5/2
= -√5
Therefore, wy(2, -4, 2) = -√5.
3. wz denotes the partial derivative of w with respect to z.
To find wz, we differentiate the function sqrt(5x^2 + y^2 - 4z^2) with respect to z:
wz = (1/2)(5x^2 + y^2 - 4z^2)^(-1/2) * d/dz (5x^2 + y^2 - 4z^2)
Now, let's differentiate the expression inside the parentheses with respect to z:
d/dz (5x^2 + y^2 - 4z^2) = -8z.
Substituting the given point (2, -4, 2) into wz:
wz(2, -4, 2) = (1/2)(5(2)^2 + (-4)^2 - 4(2)^2)^(-1/2) * (-8)(2)
= (1/2)(20 + 16 - 16)^(-1/2) * (-16)
= (1/2)(20)^(-1/2) * (-16)
= (1/2)(1/√20) * (-16)
= -16/√20
= (-16/√20) * (√20/√20)
= -16√20/20
= -4√5/5
= -√5/5
Therefore, wz(2, -4, 2) = -√5/5.
To summarize:
wx(2, -4, 2) = √5
wy(2, -4, 2) = -√5
wz(2, -4, 2) = -√5/5