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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How to Evaluate Composite Functions

from Math 103: Precalculus

Chapter 8 / Lesson 5
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