Identify Functions: To find the derivative of g(x) = ln(cos(x^3)), we will use the chain rule, which states that the derivative of a composite function is the derivative of the outer function evaluated at the inner function times the derivative of the inner function.Derivative of Outer Function: First, let's identify the outer function and the inner function. The outer function is ln(u), where u is the inner function. In this case, the inner function is cos(x^3).Derivative of Inner Function: The derivative of the outer function ln(u) with respect to u is 1/u. So, when we take the derivative of ln(cos(x^3)), we will have 1/cos(x^3) times the derivative of the inner function cos(x^3).Combine Derivatives: Now, we need to find the derivative of the inner function cos(x^3). The derivative of cos(u) with respect to u is -sin(u). Therefore, the derivative of cos(x^3) with respect to x is -sin(x^3) times the derivative of x^3 with respect to x.Simplify Expression: The derivative of x^3 with respect to x is 3x^2. So, the derivative of cos(x^3) with respect to x is -sin(x^3) * 3x^2.Final Derivative: Putting it all together, the derivative of g(x) = ln(cos(x^3)) is the derivative of the outer function times the derivative of the inner function, which is (1/cos(x^3)) * (-sin(x^3) * 3x^2).Simplifying the expression, we get g'(x) = -3x^2 * sin(x^3) / cos(x^3).We can also express sin(x^3)/cos(x^3) as tan(x^3), so the final simplified form of the derivative is g'(x) = -3x^2 * tan(x^3).

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$3$ . $g(x)=\ln \left(\cos \left(x^{3}\right)\right)$

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Find derivatives of logarithmic functions

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g(x)=\ln \left(\cos \left(x^{3}\right)\right)

Identify Functions: To find the derivative of $g(x) = \ln(\cos(x^3))$ , we will use the chain rule, which states that the derivative of a composite function is the derivative of the outer function evaluated at the inner function times the derivative of the inner function.
Derivative of Outer Function: First, let's identify the outer function and the inner function. The outer function is $\ln(u)$ , where $u$ is the inner function. In this case, the inner function is $\cos(x^3)$ .
Derivative of Inner Function: The derivative of the outer function $\ln(u)$ with respect to $u$ is $\frac{1}{u}$ . So, when we take the derivative of $\ln(\cos(x^3))$ , we will have $\frac{1}{\cos(x^3)}$ times the derivative of the inner function $\cos(x^3)$ .
Combine Derivatives: Now, we need to find the derivative of the inner function $\cos(x^3)$ . The derivative of $\cos(u)$ with respect to $u$ is $-\sin(u)$ . Therefore, the derivative of $\cos(x^3)$ with respect to $x$ is $-\sin(x^3)$ times the derivative of $x^3$ with respect to $x$ .
Simplify Expression: The derivative of $x^3$ with respect to $x$ is $3x^2$ . So, the derivative of $\cos(x^3)$ with respect to $x$ is $-\sin(x^3) \cdot 3x^2$ .
Final Derivative: Putting it all together, the derivative of $g(x) = \ln(\cos(x^3))$ is the derivative of the outer function times the derivative of the inner function, which is $(1/\cos(x^3)) \cdot (-\sin(x^3) \cdot 3x^2)$ .
Final Derivative: Putting it all together, the derivative of $g(x) = \ln(\cos(x^3))$ is the derivative of the outer function times the derivative of the inner function, which is $(1/\cos(x^3)) * (-\sin(x^3) * 3x^2)$ . Simplifying the expression, we get $g'(x) = -3x^2 * \sin(x^3) / \cos(x^3)$ .
Final Derivative: Putting it all together, the derivative of $g(x) = \ln(\cos(x^3))$ is the derivative of the outer function times the derivative of the inner function, which is $(1/\cos(x^3)) * (-\sin(x^3) * 3x^2)$ .Simplifying the expression, we get $g'(x) = -3x^2 * \sin(x^3) / \cos(x^3)$ .We can also express $\sin(x^3)/\cos(x^3)$ as $\tan(x^3)$ , so the final simplified form of the derivative is $g'(x) = -3x^2 * \tan(x^3)$ .