find the total differential: u=sqrt(x^2+3y^2)
du = du/dx * dx + du/dy * dy
u= (x^2+3y^2)^0.5
du = .5 (x^2+3y^2)^-.5 * 2 x dx + .5 (x^2+3y^2)^-.5 * 6 y dy
du = [ x dx + 3 y dy ] / sqrt(x^2+3y^2)
Oh, I see you want to dive into some calculus! Don't worry, I won't run away from differentiation. Now, let's find the total differential of u = √(x^2 + 3y^2).
To find the total differential, we need to differentiate u with respect to both variables, x and y. Here's how we do it:
∂u/∂x = (1/2)*(x^2 + 3y^2)^(-1/2)*(2x)
∂u/∂y = (1/2)*(x^2 + 3y^2)^(-1/2)*(6y)
Now, put it all together:
du = (∂u/∂x)*dx + (∂u/∂y)*dy
Plugging in the partial derivatives we found earlier:
du = (1/2)*(x^2 + 3y^2)^(-1/2)*(2x)*dx + (1/2)*(x^2 + 3y^2)^(-1/2)*(6y)*dy
And there you have it, the total differential of u = √(x^2 + 3y^2)!
If you have any more calculus questions, feel free to ask!
To find the total differential of the function u = √(x^2 + 3y^2), we need to find the partial derivatives with respect to both x and y, and then multiply them by dx and dy, respectively. The total differential can be represented as follows:
du = (∂u/∂x)dx + (∂u/∂y)dy
Let's find the partial derivatives:
∂u/∂x: To find (∂u/∂x), we'll differentiate u with respect to x while treating y as a constant.
Differentiating u = √(x^2 + 3y^2) with respect to x:
∂u/∂x = (1/2) * (2x) / √(x^2 + 3y^2)
= x / √(x^2 + 3y^2)
∂u/∂y: To find (∂u/∂y), we'll differentiate u with respect to y while treating x as a constant.
Differentiating u = √(x^2 + 3y^2) with respect to y:
∂u/∂y = (1/2) * (2 * 3y) / √(x^2 + 3y^2)
= 3y / √(x^2 + 3y^2)
Now, we multiply each partial derivative with dx and dy, respectively:
du = (∂u/∂x)dx + (∂u/∂y)dy
= (x / √(x^2 + 3y^2))dx + (3y / √(x^2 + 3y^2))dy
So, the total differential of u = √(x^2 + 3y^2) is:
du = (x / √(x^2 + 3y^2))dx + (3y / √(x^2 + 3y^2))dy
To find the total differential of the function u = √(x^2 + 3y^2), we can use the concept of partial derivatives. The total differential is determined by calculating the derivative of the function with respect to each variable and then multiplying each derivative by its corresponding differential.
Let's calculate the total differential step by step:
1. Calculate the partial derivative of u with respect to x (denoted as ∂u/∂x):
To find ∂u/∂x, we differentiate u with respect to x while treating y as a constant.
∂u/∂x = d(u)/d(x) = (1/2)*(x^2 + 3y^2)^(-1/2) * 2x = x/(√(x^2 + 3y^2))
2. Calculate the partial derivative of u with respect to y (denoted as ∂u/∂y):
To find ∂u/∂y, we differentiate u with respect to y while treating x as a constant.
∂u/∂y = d(u)/d(y) = (1/2)*(x^2 + 3y^2)^(-1/2) * 6y = 3y/(√(x^2 + 3y^2))
3. Express the total differential (du) in terms of differentials of x and y:
du = (∂u/∂x) * dx + (∂u/∂y) * dy
substituting the partial derivatives calculated above:
du = (x/(√(x^2 + 3y^2))) * dx + (3y/(√(x^2 + 3y^2))) * dy
Therefore, the total differential of the function u = √(x^2 + 3y^2) is given by:
du = (x/(√(x^2 + 3y^2))) * dx + (3y/(√(x^2 + 3y^2))) * dy