Please help me slove this and chose the correct answer. ### Calculating the Derivative of the Product of Two FunctionsLet $\alpha$ be a real valued constant.Consider the functions:- $ f(x) = 3\alpha x^4 $- $ h(x) = \sqrt{2\alpha x + 5} $We are asked to determine the derivative of the product $ f(x) \cdot h(x) $, denoted as $\frac{d}{dx}\left(f(x) h(x)\right)$.The expression for the derivative is represented as follows:\[ \frac{d}{dx} \left(f(x) h(x)\right) = \]We need to choose the correct expression from the given options.### Options:1. $\frac{3\alpha^3(9\alpha x + 20)\sqrt{2\alpha x + 5}}{2\alpha x + 5}$2. $\frac{\left(27\alpha x^4 + 12x^3\right) (2\alpha x + 5)}{\sqrt{2\alpha x + 5}}$3. $\frac{12\alpha x^3 \sqrt{2\alpha x + 5}}{2\alpha x + 5}$4. $\frac{12 \alpha x^4 + 12x^3 + 3\alpha^2 x \sqrt{2\alpha x + 5}}{\sqrt{2\alpha x + 5}}$To find the correct expression, we use the product rule for differentiation:\[ \frac{d}{dx} [f(x) \cdot h(x)] = f'(x)h(x) + f(x)h'(x) \]Calculate the derivatives $ f'(x) $ and $ h'(x) $:$ f'(x) = \frac{d}{dx} (3\alpha x^4) = 12\alpha x^3 $Using the chain rule for $ h(x) $:\[ h(x) = (2\alpha x + 5)^{1/2} \]\[ h'(x) = \frac{1}{2}(2\alpha x + 5)^{-1/2} \cdot 2\alpha \]\[ h'(x) = \frac{\alpha}{\sqrt{2\

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Description: Please help me slove this and chose the correct answer. ### Calculating the Derivative of the Product of Two FunctionsLet $\alpha$ be a real valued constant.Consider the functions:- $ f(x) = 3\alpha x^4 $- $ h(x) = \sqrt{2\alpha x + 5} $We are

Here we will use chain rule of differentiation to find the solution

Answered: Let a be a real valued constant. Let f…

Math

Calculus

Let a be a real valued constant. Let f (x) = 3axt. Let h (x) = /2ax + 5 . Determine (f (x) h (x)) . S (x) h (x)) dx

Calculus: Early Transcendentals

8th Edition

ISBN:9781285741550

Author:James Stewart

Publisher:James Stewart

Chapter1: Functions And Models

Section: Chapter Questions

Problem 1RCC: (a) What is a function? What are its domain and range? (b) What is the graph of a function? (c) How...

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Please help me slove this and chose the correct answer.

### Calculating the Derivative of the Product of Two Functions

Let $\alpha$ be a real valued constant.

Consider the functions:
- $ f(x) = 3\alpha x^4 $
- $ h(x) = \sqrt{2\alpha x + 5} $

We are asked to determine the derivative of the product $ f(x) \cdot h(x) $, denoted as $\frac{d}{dx}\left(f(x) h(x)\right)$.

The expression for the derivative is represented as follows:
\[ \frac{d}{dx} \left(f(x) h(x)\right) = \]
We need to choose the correct expression from the given options.

### Options:

1. $\frac{3\alpha^3(9\alpha x + 20)\sqrt{2\alpha x + 5}}{2\alpha x + 5}$
2. $\frac{\left(27\alpha x^4 + 12x^3\right) (2\alpha x + 5)}{\sqrt{2\alpha x + 5}}$
3. $\frac{12\alpha x^3 \sqrt{2\alpha x + 5}}{2\alpha x + 5}$
4. $\frac{12 \alpha x^4 + 12x^3 + 3\alpha^2 x \sqrt{2\alpha x + 5}}{\sqrt{2\alpha x + 5}}$

To find the correct expression, we use the product rule for differentiation:
\[ \frac{d}{dx} [f(x) \cdot h(x)] = f'(x)h(x) + f(x)h'(x) \]

Calculate the derivatives $ f'(x) $ and $ h'(x) $:
$ f'(x) = \frac{d}{dx} (3\alpha x^4) = 12\alpha x^3 $

Using the chain rule for $ h(x) $:
\[ h(x) = (2\alpha x + 5)^{1/2} \]
\[ h'(x) = \frac{1}{2}(2\alpha x + 5)^{-1/2} \cdot 2\alpha \]
\[ h'(x) = \frac{\alpha}{\sqrt{2\

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