Let z=2x+5y−−−−−−√
. Then:
The rate of change in z
at (2,4)
as we change x
but hold y
fixed is
, and
The rate of change in z
at (2,4)
as we change y
but hold x
fixed is
To find the rate of change in z at (2,4) as we change x but hold y fixed, we need to find the partial derivative of z with respect to x.
First, let's differentiate each term in z with respect to x:
∂z/∂x = ∂(2x)/∂x + ∂(5y)/∂x - ∂(√)/∂x
The derivative of 2x with respect to x is simply 2.
The derivative of 5y with respect to x is 0, since y is held fixed and does not vary with x.
The derivative of √( ) with respect to x can be found by applying the chain rule. Let's define a new function u = 2x + 5y. Then we have:
∂(√(u))/∂x = ∂(√(u))/∂u * ∂u/∂x
The derivative of √(u) with respect to u is 1/(2√(u)).
The derivative of u with respect to x is 2.
Substituting these values back into our partial derivative expression, we have:
∂z/∂x = 2 + 0 - (1/(2√(u))) * 2
At the point (2,4), u = 2(2) + 5(4) = 8 + 20 = 28.
√(28) can be simplified as 2√(7).
So, ∂z/∂x at (2,4) is:
∂z/∂x = 2 - (1/(2√(7))) * 2
To find the rate of change in z at (2,4) as we change y but hold x fixed, we need to find the partial derivative of z with respect to y.
Using a similar process as above, we have:
∂z/∂y = ∂(2x)/∂y + ∂(5y)/∂y - ∂(√)/∂y
The derivative of 2x with respect to y is 0, since x is held fixed and does not vary with y.
The derivative of 5y with respect to y is 5.
The derivative of √(u) with respect to y can be found using the chain rule. We already defined u = 2x + 5y. In this case, we have:
∂(√(u))/∂y = ∂(√(u))/∂u * ∂u/∂y
The derivative of √(u) with respect to u is also 1/(2√(u)).
The derivative of u with respect to y is 5.
Substituting these values back into our partial derivative expression, we have:
∂z/∂y = 0 + 5 - (1/(2√(u))) * 5
At the point (2,4), u = 28.
So, ∂z/∂y at (2,4) is:
∂z/∂y = 5 - (1/(2√(28))) * 5
To find the rate of change in z at (2,4) as we change x but hold y fixed, we need to calculate the derivative of z with respect to x.
Step 1: Find the derivative of z with respect to x:
Differentiate each term of z with respect to x while treating y as a constant.
d/dx(z) = d/dx(2x) + d/dx(5y) - d/dx(sqrt())
Step 2: Simplify the derivative:
Since d/dx(2x) = 2 and d/dx(5y) = 0 (since y is held fixed), we only need to evaluate the derivative of the square root term.
d/dx(sqrt()) = 1 / (2 * sqrt())
Step 3: Substitute the values of x=2 and y=4 into the derivative:
Substituting x=2 and y=4 into d/dx(z), we get:
d/dx(z) = 2 + 0 - 1 / (2 * sqrt()) = 2 - 1 / (2 * sqrt())
So, the rate of change in z at (2,4) as we change x but hold y fixed is 2 - 1 / (2 * sqrt()).
To find the rate of change in z at (2,4) as we change y but hold x fixed, we need to calculate the derivative of z with respect to y.
Step 1: Find the derivative of z with respect to y:
Differentiate each term of z with respect to y while treating x as a constant.
d/dy(z) = d/dy(2x) + d/dy(5y) - d/dy(sqrt())
Step 2: Simplify the derivative:
Since d/dy(2x) = 0 (since x is held fixed), d/dy(5y) = 5, and d/dy(sqrt()) = 0 (since y is held fixed), the derivative simplifies to:
d/dy(z) = 0 + 5 - 0 = 5
So, the rate of change in z at (2,4) as we change y but hold x fixed is 5.