Find the partial derivative for x.
f(x,y)=1/SQRTx^2+y^2
treat y as a constant:
f(x,y) = (x^2 + y^2)^.5
dy/dx = .5 (x^2 + y^2)^-.5 * (2x)
= x/sqrt(x^2 + y^2)
To find the partial derivative of f(x, y) with respect to x, we treat y as a constant and differentiate f(x, y) with respect to x.
Let's start by rewriting the function:
f(x, y) = (x^2 + y^2)^(-1/2)
Now, let's find the partial derivative of f(x, y) with respect to x. We differentiate each term with respect to x while treating y as a constant:
∂f/∂x = ∂/∂x (x^2 + y^2)^(-1/2)
To differentiate this, let's apply the chain rule. The chain rule states that if we have a composition of functions, f(g(x)), then the derivative of f(g(x)) with respect to x is given by f'(g(x)) * g'(x).
Let u = x^2 + y^2, so that f(x, y) = u^(-1/2). Applying the chain rule:
∂f/∂x = ∂/∂x (u^(-1/2))
= (-1/2) * u^(-3/2) * ∂u/∂x
Now, let's find ∂u/∂x:
∂u/∂x = ∂/∂x (x^2 + y^2)
= 2x
Let's substitute this back into our previous expression:
∂f/∂x = (-1/2) * u^(-3/2) * ∂u/∂x
= (-1/2) * (x^2 + y^2)^(-3/2) * (2x)
Simplifying this expression, we get:
∂f/∂x = -x * (x^2 + y^2)^(-3/2)
Therefore, the partial derivative of f(x, y) with respect to x is -x * (x^2 + y^2)^(-3/2).
To find the partial derivative of f(x,y) with respect to x, we need to differentiate f(x,y) with respect to x while treating y as a constant.
Step 1: Rewrite the function
f(x,y) = (x^2 + y^2)^(-1/2)
Step 2: Apply the chain rule
To differentiate f(x,y) with respect to x, we first differentiate the outer function with respect to the inner function and then multiply it by the derivative of the inner function with respect to x.
The derivative of the outer function (u^(-1/2)) with respect to u is:
d/dx (u^(-1/2)) = (-1/2)u^(-1/2-1) = -1/2u^(3/2)
The inner function u in this case is x^2 + y^2.
The derivative of the inner function (x^2 + y^2) with respect to x is:
d/dx (x^2 + y^2) = 2x
Step 3: Combine the derivatives
Now, multiply the derivative of the outer function with the derivative of the inner function:
-1/2(x^2 + y^2)^(3/2) * 2x
So, the partial derivative of f(x,y) with respect to x is:
∂f/∂x = -x(x^2 + y^2)^(-3/2)