How to Differentiate the Square Root Function f(x) = √(1 – x)?

To differentiate the square root function f(x) = √(1 – x), we can apply the chain rule of differentiation. The chain rule states that if you have a composite function, the derivative is the derivative of the outer function evaluated at the inner function multiplied by the derivative of the inner function.

In this case, let’s rewrite the function in a more manageable form:

f(x) = (1 – x)^(1/2)

Now, we can differentiate it:

1. Identify the outer function and inner function:
   • Outer function: u^(1/2)
   • Inner function: u = 1 – x
2. Differentiate the outer function:
   • The derivative of u^(1/2) is (1/2)u^(-1/2).
3. Differentiate the inner function:
   • The derivative of 1 – x is -1.
4. Apply the chain rule:
• f'(x) = (1/2)(1 – x)^(-1/2) * (-1)
5. Simplify the expression:
• f'(x) = -1/(2√(1 – x))

So, the derivative of the square root function f(x) = √(1 – x) is:

f'(x) = -1 / (2√(1 – x))

This result tells us how the function f(x) changes as x changes. The negative sign indicates that as x increases, the value of f(x) decreases, which aligns with the behavior of the square root function when we have a linear term subtracted from 1.