Solved on Feb 12, 2024

Find the derivative $f'(x)$ of $f(x) = x^2 \ln(12 - 4x^2)$ and its domain.

STEP 1

Assumptions
1. We are given the derivative of a function $f(x)$ , denoted as $f'(x)$ .
2. We need to find the domain of $f'(x)$ .
3. The domain of a function is the set of all possible input values (x-values) for which the function is defined.

STEP 2

Identify the components of $f'(x)$ that could restrict the domain.
$f^{\prime}(x)=2 x\left(\ln \left(12-4 x^{2}\right)\right)+x^{2}\left(\left(\frac{1}{12-4 x^{2}}\right)(-8 x)\right)$
The components that could restrict the domain are:
1. The logarithmic function $\ln(12-4x^2)$
2. The denominator in the fraction $\frac{1}{12-4x^2}$

STEP 3

Determine the domain restrictions for the logarithmic function.
The argument of a logarithmic function must be greater than zero.
$12-4x^2 > 0$

STEP 4

Solve the inequality to find the domain restrictions for the logarithmic function.
$12 > 4x^2$

STEP 5

Divide both sides of the inequality by 4.
$\frac{12}{4} > x^2$

STEP 6

Simplify the inequality.
$3 > x^2$

STEP 7

Take the square root of both sides, remembering to consider both the positive and negative roots.
$\sqrt{3} > x \quad \text{and} \quad -\sqrt{3} < x$

STEP 8

Combine the two inequalities to express the domain restriction for the logarithmic function.
$- \sqrt{3} < x < \sqrt{3}$

STEP 9

Determine the domain restrictions for the fraction $\frac{1}{12-4x^2}$ .
The denominator cannot be zero, so we set it not equal to zero.
$12-4x^2 \neq 0$

STEP 10

Solve the equation to find the values that are not in the domain.
$4x^2 \neq 12$

STEP 11

Divide both sides of the equation by 4.
$x^2 \neq 3$

STEP 12

Take the square root of both sides, remembering to consider both the positive and negative roots.
$x \neq \sqrt{3} \quad \text{and} \quad x \neq -\sqrt{3}$

STEP 13

Combine the domain restrictions from the logarithmic function and the fraction.
The domain of $f'(x)$ is all real numbers except $x = \sqrt{3}$ and $x = -\sqrt{3}$ , and within the interval $(-\sqrt{3}, \sqrt{3})$ .

STEP 14

Write the final domain of $f'(x)$ .
The domain of $f'(x)$ is:
$\text{Domain} = \{ x \in \mathbb{R} \mid -\sqrt{3} < x < \sqrt{3} \}$

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Find the derivative $f'(x)$ of $f(x) = x^2 \ln(12 - 4x^2)$ and its domain.

STEP 1

Assumptions
1. We are given the derivative of a function $f(x)$ , denoted as $f'(x)$ .
2. We need to find the domain of $f'(x)$ .
3. The domain of a function is the set of all possible input values (x-values) for which the function is defined.

STEP 2

STEP 3

Determine the domain restrictions for the logarithmic function.
The argument of a logarithmic function must be greater than zero.
$12-4x^2 > 0$

STEP 4

Solve the inequality to find the domain restrictions for the logarithmic function.
$12 > 4x^2$

STEP 5

Divide both sides of the inequality by 4.
$\frac{12}{4} > x^2$

STEP 6

Simplify the inequality.
$3 > x^2$

STEP 7

Take the square root of both sides, remembering to consider both the positive and negative roots.
$\sqrt{3} > x \quad \text{and} \quad -\sqrt{3} < x$

STEP 8

Combine the two inequalities to express the domain restriction for the logarithmic function.
$- \sqrt{3} < x < \sqrt{3}$

STEP 9

Determine the domain restrictions for the fraction $\frac{1}{12-4x^2}$ .
The denominator cannot be zero, so we set it not equal to zero.
$12-4x^2 \neq 0$

STEP 10

Solve the equation to find the values that are not in the domain.
$4x^2 \neq 12$

STEP 11

Divide both sides of the equation by 4.
$x^2 \neq 3$

STEP 12

Take the square root of both sides, remembering to consider both the positive and negative roots.
$x \neq \sqrt{3} \quad \text{and} \quad x \neq -\sqrt{3}$

STEP 13

Combine the domain restrictions from the logarithmic function and the fraction.
The domain of $f'(x)$ is all real numbers except $x = \sqrt{3}$ and $x = -\sqrt{3}$ , and within the interval $(-\sqrt{3}, \sqrt{3})$ .

STEP 14

Write the final domain of $f'(x)$ .
The domain of $f'(x)$ is:
$\text{Domain} = \{ x \in \mathbb{R} \mid -\sqrt{3} < x < \sqrt{3} \}$

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Find the derivative f′(x)f'(x)f′(x) of f(x)=x2ln⁡(12−4x2)f(x) = x^2 \ln(12 - 4x^2)f(x)=x2ln(12−4x2) and its domain.

STEP 1

Assumptions1. We are given the derivative of a function f(x)f(x)f(x), denoted as f′(x)f'(x)f′(x).2. We need to find the domain of f′(x)f'(x)f′(x).3. The domain of a function is the set of all possible input values (x-values) for which the function is defined.

STEP 2

STEP 3

Determine the domain restrictions for the logarithmic function.The argument of a logarithmic function must be greater than zero.12−4x2>012-4x^2 > 012−4x2>0

STEP 4

Solve the inequality to find the domain restrictions for the logarithmic function.12>4x212 > 4x^212>4x2

STEP 5

Divide both sides of the inequality by 4.124>x2\frac{12}{4} > x^2412​>x2

STEP 6

Simplify the inequality.3>x23 > x^23>x2

STEP 7

Take the square root of both sides, remembering to consider both the positive and negative roots.3>xand−3<x\sqrt{3} > x \quad \text{and} \quad -\sqrt{3} < x3​>xand−3​<x

STEP 8

Combine the two inequalities to express the domain restriction for the logarithmic function.−3<x<3- \sqrt{3} < x < \sqrt{3}−3​<x<3​

STEP 9

Determine the domain restrictions for the fraction 112−4x2\frac{1}{12-4x^2}12−4x21​.The denominator cannot be zero, so we set it not equal to zero.12−4x2≠012-4x^2 \neq 012−4x2=0

STEP 10

Solve the equation to find the values that are not in the domain.4x2≠124x^2 \neq 124x2=12

STEP 11

Divide both sides of the equation by 4.x2≠3x^2 \neq 3x2=3

STEP 12

Take the square root of both sides, remembering to consider both the positive and negative roots.x≠3andx≠−3x \neq \sqrt{3} \quad \text{and} \quad x \neq -\sqrt{3}x=3​andx=−3​

STEP 13

Combine the domain restrictions from the logarithmic function and the fraction.The domain of f′(x)f'(x)f′(x) is all real numbers except x=3x = \sqrt{3}x=3​ and x=−3x = -\sqrt{3}x=−3​, and within the interval (−3,3)(-\sqrt{3}, \sqrt{3})(−3​,3​).

STEP 14

Write the final domain of f′(x)f'(x)f′(x).The domain of f′(x)f'(x)f′(x) is:Domain={x∈R∣−3<x<3}\text{Domain} = \{ x \in \mathbb{R} \mid -\sqrt{3} < x < \sqrt{3} \}Domain={x∈R∣−3​<x<3​}

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Find the derivative $f'(x)$ of $f(x) = x^2 \ln(12 - 4x^2)$ and its domain.

Assumptions
1. We are given the derivative of a function $f(x)$ , denoted as $f'(x)$ .
2. We need to find the domain of $f'(x)$ .
3. The domain of a function is the set of all possible input values (x-values) for which the function is defined.

Determine the domain restrictions for the logarithmic function.
The argument of a logarithmic function must be greater than zero.
$12-4x^2 > 0$

Solve the inequality to find the domain restrictions for the logarithmic function.
$12 > 4x^2$

Divide both sides of the inequality by 4.
$\frac{12}{4} > x^2$

Simplify the inequality.
$3 > x^2$

Take the square root of both sides, remembering to consider both the positive and negative roots.
$\sqrt{3} > x \quad \text{and} \quad -\sqrt{3} < x$

Combine the two inequalities to express the domain restriction for the logarithmic function.
$- \sqrt{3} < x < \sqrt{3}$

Determine the domain restrictions for the fraction $\frac{1}{12-4x^2}$ .
The denominator cannot be zero, so we set it not equal to zero.
$12-4x^2 \neq 0$

Solve the equation to find the values that are not in the domain.
$4x^2 \neq 12$

Divide both sides of the equation by 4.
$x^2 \neq 3$

Take the square root of both sides, remembering to consider both the positive and negative roots.
$x \neq \sqrt{3} \quad \text{and} \quad x \neq -\sqrt{3}$

Combine the domain restrictions from the logarithmic function and the fraction.
The domain of $f'(x)$ is all real numbers except $x = \sqrt{3}$ and $x = -\sqrt{3}$ , and within the interval $(-\sqrt{3}, \sqrt{3})$ .

Write the final domain of $f'(x)$ .
The domain of $f'(x)$ is:
$\text{Domain} = \{ x \in \mathbb{R} \mid -\sqrt{3} < x < \sqrt{3} \}$