# We are given that y = f(x) with f(1) = 5 and f'(1) = 7. a. Find g'(1) if g ( x ) = \sqrt f...

## Question:

We are given that y = f(x) with f(1) = 5 and f'(1) = 7.

a. Find g'(1) if {eq}g\left ( x \right ) \ = \ \sqrt{f\left ( x \right )} {/eq}.

b. Find h'(1) if {eq}h\left ( x \right ) \ = \ f\left (\sqrt{x} \right ) {/eq}

## Derivative of a Composite Function:

Normally, when we find the derivative of a function {eq}f(x) {/eq} then its derivative with respect to {eq}x \,\, is \,\, f^{'} (x) {/eq}

Now, if the function is in Composite form that is {eq}f(g(x)) {/eq} then its derivative will be as follows as:

$$\frac{d}{dx} (f(g(x)) = f^{'} (g(x)) \frac{d}{dx} (g(x)) $$

## Answer and Explanation:

We are given:

{eq}f(1) = 5 \,\, and \,\, f^{'} (1) = 7 {/eq}

a.)

{eq}g(x) = \sqrt{ f(x)} = \left( f(x) \right)^{\frac{1}{2}} {/eq}

Derivative of...

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View this answerWe are given:

{eq}f(1) = 5 \,\, and \,\, f^{'} (1) = 7 {/eq}

a.)

{eq}g(x) = \sqrt{ f(x)} = \left( f(x) \right)^{\frac{1}{2}} {/eq}

Derivative of {eq}g(x) {/eq} with respect to {eq}x {/eq} is:

{eq}\begin{align*} g^{'}(x) &= \frac{d}{dx} \left[ \left( f(x) \right)^{\frac{1}{2}} \right] \\ &= \frac{1}{2} \left( f(x) \right)^{( \frac{1}{2} - 1)} \frac{d}{dx} (f(x)) \hspace{1 cm} \left[ \because \frac{d}{dx} (x^n) = nx^{(n - 1)} \right] \\ &= \frac{1}{2} \left( f(x) \right)^{ -\frac{1}{2}} f^{'} (x) \\ &= \frac{f^{'}(x)}{2 \sqrt{ f(x)}} \end{align*} {/eq}

Now:

{eq}x = 1 {/eq}

{eq}\begin{align*} g^{'} (1) &= \frac{f^{'} (1)}{2 \sqrt{ f(1)}} \\ &= \frac{7}{2 \sqrt{5}} \hspace{1 cm} \left[ \because f(1) = 5 \,\, and \,\, f^{'} (1) = 7 \right] \end{align*} {/eq}

b.)

{eq}h(x) = f( \sqrt{x}) {/eq}

Derivative of {eq}h(x) {/eq} with respect to {eq}x {/eq} is:

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