Find the derivative of y=sqrt(1+sqrtx)

Identify Function: Identify the function to differentiate. We are given the function y = sqrt(1 + sqrt(x)). We need to find its derivative with respect to x.Apply Chain Rule: Apply the chain rule for differentiation. 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. In this case, we have an outer function which is the square root and an inner function which is (1 + sqrt(x)).Differentiate Outer Function: Differentiate the outer function. The outer function is sqrt(u), where u = 1 + sqrt(x). The derivative of sqrt(u) with respect to u is 1/(2*sqrt(u)). We will substitute back for u later.Differentiate Inner Function: Differentiate the inner function. The inner function is u = 1 + sqrt(x). The derivative of u with respect to x is the derivative of sqrt(x) with respect to x, which is 1/(2*sqrt(x)).Apply Chain Rule Multiplication: Apply the chain rule by multiplying the derivatives of the outer and inner functions. The derivative of y with respect to x is (1/(2*sqrt(1 + sqrt(x)))) * (1/(2*sqrt(x))).Simplify Expression: Simplify the expression. We can leave the derivative as it is or simplify further, but for the sake of this problem, we will leave it in its current state.

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Find the derivative of $\newline$ $y=\sqrt{1+\sqrt{x}}$

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Algebra 1

Domain and range of square root functions: equations

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Q. Find the derivative of

\newline

y=\sqrt{1+\sqrt{x}}

Identify Function: Identify the function to differentiate. $\newline$ We are given the function $y = \sqrt{1 + \sqrt{x}}$ . We need to find its derivative with respect to $x$ .
Apply Chain Rule: Apply the chain rule for differentiation. 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. In this case, we have an outer function which is the square root and an inner function which is $(1 + \sqrt{x})$ .
Differentiate Outer Function: Differentiate the outer function. $\newline$ The outer function is $\sqrt{u}$ , where $u = 1 + \sqrt{x}$ . The derivative of $\sqrt{u}$ with respect to $u$ is $\frac{1}{2\sqrt{u}}$ . We will substitute back for $u$ later.
Differentiate Inner Function: Differentiate the inner function. $\newline$ The inner function is $u = 1 + \sqrt{x}$ . The derivative of $u$ with respect to $x$ is the derivative of $\sqrt{x}$ with respect to $x$ , which is $\frac{1}{2\sqrt{x}}$ .
Apply Chain Rule Multiplication: Apply the chain rule by multiplying the derivatives of the outer and inner functions. $\newline$ The derivative of $y$ with respect to $x$ is $\frac{1}{2\sqrt{1 + \sqrt{x}}} \cdot \frac{1}{2\sqrt{x}}$ .
Simplify Expression: Simplify the expression. $\newline$ We can leave the derivative as it is or simplify further, but for the sake of this problem, we will leave it in its current state.