Find the derivative of f(x). f(x)=sqrt(x-1)

Apply Chain Rule: To find the derivative of f(x) = sqrt(x-1), we need to apply the chain rule. The chain rule states that the derivative of a composite function is the derivative of the outer function evaluated at the inner function times the derivative of the inner function.Identify Outer Function: The outer function is the square root function, which can be written as (x-1)^(1/2). The derivative of x^(1/2) with respect to x is (1/2)x^(-1/2).Identify Inner Function: The inner function is (x-1). The derivative of (x-1) with respect to x is 1, since the derivative of a constant is 0 and the derivative of x is 1.Apply Chain Rule Again: Applying the chain rule, we multiply the derivative of the outer function by the derivative of the inner function. This gives us: f'(x) = (1/2)(x-1)^(-1/2) * 1Simplify Final Derivative: Simplify the expression to get the final derivative: f'(x) = 1 / (2sqrt(x-1))

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Q. Find the derivative of

f(x)

\newline

f(x)=\sqrt{x-1}

Apply Chain Rule: To find the derivative of $f(x) = \sqrt{x-1}$ , we need to apply the chain rule. The chain rule states that the derivative of a composite function is the derivative of the outer function evaluated at the inner function times the derivative of the inner function.
Identify Outer Function: The outer function is the square root function, which can be written as x-1)^{\frac{1}{2}}\. The derivative of \$x^{\frac{1}{2}} with respect to $x$ is $\frac{1}{2}x^{-\frac{1}{2}}$ .
Identify Inner Function: The inner function is $(x-1)$ . The derivative of $(x-1)$ with respect to $x$ is $1$ , since the derivative of a constant is $0$ and the derivative of $x$ is $1$ .
Apply Chain Rule Again: Applying the chain rule, we multiply the derivative of the outer function by the derivative of the inner function. This gives us: $\newline$ $f'(x) = (\frac{1}{2})(x-1)^{-\frac{1}{2}} \times 1$
Simplify Final Derivative: Simplify the expression to get the final derivative: $f'(x) = \frac{1}{2\sqrt{x-1}}$