Chapter 52: Chapter 52

Problem 971

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Find the derivative of: $\left.g(x)=x^{3} /\left[{ }^{2} \sqrt{(} 3 x^{2}-1\right)\right]$.

Short Answer

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The derivative of $g(x) = \frac{x^3}{2\sqrt{3x^2 - 1}}$ is $g'(x) = \frac{3x^2 - 2x^3}{4(3x^2 - 1)}$.

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TABLE OF CONTENTS

Step 1: Identify the outside and inside function

In the given function, $ g(x) = \frac{x^3}{2 \sqrt{3x^2 - 1}} $, the outside function is the division operation of $ x^3 $ by the square root part, and the inside function is $ 3x^2 - 1 $ under the square root.

Step 2: Apply the Quotient Rule

The quotient rule states that the derivative of $\frac{u}{v}$ is $\frac{vu' - uv'}{v^2}$, where $u'$ is the derivative of u and $v'$ is the derivative of v. Here, $u = x^3$ and $v = 2\sqrt{3x^2 - 1}$. First, find the derivatives of $u$ and $v$: - Derivative of $u = x^3$ is $u' = 3x^2$ using the power rule. - Derivative of $v = 2\sqrt{3x^2 - 1}$ requires the chain rule since $3x^2 - 1$ is nested inside the square root function.

Step 3: Apply the Chain Rule for $v'$

The derivative of the function $v = 2\sqrt{3x^2 - 1}$ is $v' = 2 \frac{1}{2\sqrt{3x^2 - 1}} \cdot (6x)$. The $2$ and $\frac{1}{2}$ cancel out, and the slope is $v' = \frac{6x}{\sqrt{3x^2 - 1}}$.

Step 4: Substitute Expressions in Quotient Rule

Substitute $u', v'$ and $v$ into the quotient rule formula. Thus, the derivative of the function $g = \frac{u'}{v} = \frac{v(3x^2) - u * \frac{6x}{\sqrt{3x^2 - 1}}}{(2 \sqrt{3x^2 - 1})^2}$.

Step 5: Simplify

The function can be further simplified: - Multiply $v$ and $3x^2$ to get $6x^5 - 2x^3$ - The denominator when squared becomes $4(3x^2-1)$ Therefore, the derivative of $ \frac{x^3}{2 \sqrt{3x^2 - 1}}$ is $ \frac{3x^2 * 2\sqrt{3x^2 - 1} - x^3 * \frac{6x}{\sqrt{3x^2 - 1}}}{4(3x^2 - 1)}$. After some simplification, you get $g'(x) = \frac{3x^2 - 2x^3}{4(3x^2 -1)}$.

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Chapter 52: Chapter 52

Find the derivative of: $\left.g(x)=x^{3} /\left[{ }^{2} \sqrt{(} 3 x^{2}-1\right)\right]$.

Short Answer

Step by step solution

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Step 1: Identify the outside and inside function

Step 2: Apply the Quotient Rule

Step 3: Apply the Chain Rule for \(v'\)

Step 4: Substitute Expressions in Quotient Rule

Step 5: Simplify

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