How do I differentiate fx and fy f(x,y)=√(1+x^2 +y^2)
f = (1+x^2+y^2)^.5
df/dx = .5 (1+x^2+y^2)^-.5 * d/dx (1+x^2+y^2)
= .5 (1+x^2+y^2)^-.5 (0 + 2x + 2 y dy/dx)
= (x + y dy/dx) / √(1+x^2 +y^2)
If you're looking for partial derivatives, then as Damon showed,
Fx = x/√(1+x^2 +y^2)
Fy = y/√(1+x^2 +y^2)
To differentiate a multivariable function like f(x,y) = √(1 + x^2 + y^2), we can use partial derivatives.
The partial derivative with respect to x, denoted as ∂f/∂x or fx, measures how f changes when only x varies and y is held constant.
To find fx of f(x,y), we differentiate the function with respect to x while treating y as a constant. Here's the step-by-step process:
1. Start with the function f(x,y) = √(1 + x^2 + y^2).
2. Take the derivative of the inside function (1 + x^2 + y^2) with respect to x, which is 2x.
3. Treat y as a constant because we are differentiating with respect to x.
4. Apply the chain rule by dividing the derivative of the inside function by the derivative of the square root function, which is (1/2) * (1 + x^2 + y^2)^(-1/2).
5. Multiplying the derivative of the inside function and the chain rule results, we get fx = (2x) / (2√(1 + x^2 + y^2)).
6. Simplify the expression, and the final expression for fx is: fx = x / √(1 + x^2 + y^2).
Following the same steps, we can find the partial derivative with respect to y, denoted as ∂f/∂y or fy:
1. Start with the function f(x,y) = √(1 + x^2 + y^2).
2. Take the derivative of the inside function (1 + x^2 + y^2) with respect to y, which is 2y.
3. Treat x as a constant because we are differentiating with respect to y.
4. Apply the chain rule by dividing the derivative of the inside function by the derivative of the square root function, which is (1/2) * (1 + x^2 + y^2)^(-1/2).
5. Multiplying the derivative of the inside function and the chain rule results, we get fy = (2y) / (2√(1 + x^2 + y^2)).
6. Simplify the expression, and the final expression for fy is: fy = y / √(1 + x^2 + y^2).
So, the partial derivatives of f(x,y) = √(1 + x^2 + y^2) are fx = x / √(1 + x^2 + y^2) and fy = y / √(1 + x^2 + y^2).