"Find the derivative of f(x). f(x) = sqrt (x+3) f′(x) = ______"

Identify Functions: Identify the inner and outer functions for the composition. Inner function, u(x) = (x + 3). Outer function, f(u) = sqrt(u), where u is (x + 3).Derivative of Outer Function: Determine the derivative of the outer function, f(u) = sqrt(u). Using the power rule, the derivative of u^(1/2) is (1/2)u^(-1/2). Therefore, f'(u) = 1/(2sqrt(u)).Substitute into Derivative: Substitute u(x) = (x + 3) into the derivative of the outer function. f'(u(x)) = 1/(2sqrt(u(x))) = 1/(2sqrt(x+3)).Derivative of Inner Function: Find the derivative of the inner function, u(x) = x + 3. The derivative of x is 1, and the derivative of a constant is 0. Therefore, u'(x) = 1 + 0 = 1.Apply Chain Rule: Apply the chain rule: d/dx [f(u(x))] = f'(u(x)) * u'(x). f'(u(x)) = 1/(2sqrt(x+3)) and u'(x) = 1. Therefore, f'(x) = 1/(2sqrt(x+3)) * 1 = 1/(2sqrt(x+3)).

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Q. Find the derivative of

f(x)

\newline

f(x) = \sqrt{x+3}

\newline

f'(x) =

______

Identify Functions: Identify the inner and outer functions for the composition. $\newline$ Inner function, $u(x) = (x + 3)$ . $\newline$ Outer function, $f(u) = \sqrt{u}$ , where $u$ is $(x + 3)$ .
Derivative of Outer Function: Determine the derivative of the outer function, $f(u) = \sqrt{u}$ . $\newline$ Using the power rule, the derivative of $u^{\frac{1}{2}}$ is $\left(\frac{1}{2}\right)u^{-\frac{1}{2}}$ . $\newline$ Therefore, $f'(u) = \frac{1}{2\sqrt{u}}$ .
Substitute into Derivative: Substitute $u(x) = (x + 3)$ into the derivative of the outer function. $f'(u(x)) = \frac{1}{2\sqrt{u(x)}} = \frac{1}{2\sqrt{x+3}}.$
Derivative of Inner Function: Find the derivative of the inner function, $u(x) = x + 3$ . $\newline$ The derivative of $x$ is $1$ , and the derivative of a constant is $0$ . $\newline$ Therefore, $u'(x) = 1 + 0 = 1$ .
Apply Chain Rule: Apply the chain rule: $\frac{d}{dx} [f(u(x))] = f'(u(x)) \cdot u'(x)$ . $f'(u(x)) = \frac{1}{2\sqrt{x+3}}$ and $u'(x) = 1$ . Therefore, $f'(x) = \frac{1}{2\sqrt{x+3}} \cdot 1 = \frac{1}{2\sqrt{x+3}}$ .