What is the first derivative of

L(x) = ((4+x^2)^0.5) + ((25 + (10-x)^2)^0.5)

To find the first derivative of the function L(x) = √(4 + x^2) + √(25 + (10 - x)^2), we can use the chain rule. The chain rule states that if we have a function g(f(x)), then the derivative of g(f(x)) with respect to x is given by g'(f(x)) * f'(x).

Let's begin by finding the derivative of the first term, √(4 + x^2). We can rewrite this term as (4 + x^2)^(1/2). Applying the chain rule, the derivative of this term with respect to x is given as follows:

d/dx [(4 + x^2)^(1/2)] = (1/2)(4 + x^2)^(-1/2) * d/dx (4 + x^2).

Now, let's find the derivative of the second term, √(25 + (10 - x)^2). We can rewrite this term as (25 + (10 - x)^2)^(1/2). Again, applying the chain rule, the derivative of this term with respect to x is given as follows:

d/dx [(25 + (10 - x)^2)^(1/2)] = (1/2)(25 + (10 - x)^2)^(-1/2) * d/dx (25 + (10 - x)^2).

To find the derivative of the function L(x), we simply add the derivatives of the two terms:

L'(x) = (1/2)(4 + x^2)^(-1/2) * d/dx (4 + x^2) + (1/2)(25 + (10 - x)^2)^(-1/2) * d/dx (25 + (10 - x)^2).

Simplifying further, we can find the individual derivatives:

d/dx (4 + x^2) = 2x,

d/dx (25 + (10 - x)^2) = 2(10 - x)(-1).

Now, substituting these derivatives back into the expression, we have:

L'(x) = (1/2)(4 + x^2)^(-1/2) * 2x + (1/2)(25 + (10 - x)^2)^(-1/2) * 2(10 - x)(-1).

Simplifying this expression would give you the first derivative of the function L(x).