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a=

qquad

b=

d=
qquad
e=
qquad

qquad

qquad

qquad

qquad
c-

f=

$a=$ $\newline$ $\qquad$ $\newline$ $b=$ $\newline$ $d=$ $\qquad$ $e=$ $\qquad$ $\newline$ $\qquad$ $\newline$ $\qquad$ $\newline$ $\qquad$ $\newline$ $\qquad$ $c-$ $\newline$ $f=$

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Find derivatives of radical functions

Full solution

Q.

a=

\newline

\qquad

\newline

b=

\newline

d=

\qquad

e=

\qquad

\newline

\qquad

\newline

\qquad

\newline

\qquad

\newline

\qquad

c-

\newline

f=

Identify Function Components: Identify the function and its components. $f(x) = \sqrt{x+3}$ , where the inner function $u(x) = x + 3$ and the outer function is $\sqrt{u}$ .
Differentiate Outer Function: Differentiate the outer function with respect to $u$ . If $f(u) = \sqrt{u}$ , then $f'(u) = \frac{1}{2\sqrt{u}}$ .
Substitute Inner Function: Substitute $u(x)$ into the derivative of the outer function. $\newline$ $f'(u(x)) = \frac{1}{2\sqrt{x+3}}$ .
Differentiate Inner Function: Differentiate the inner function $u(x)$ with respect to $x$ . $u(x) = x + 3$ , so $u'(x) = 1$ .
Apply Chain Rule: Apply the chain rule to find the derivative of $f(x)$ . $f'(x) = f'(u(x)) \cdot u'(x) = \frac{1}{2\sqrt{x+3}} \cdot 1 = \frac{1}{2\sqrt{x+3}}$ .

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