10–19. Derivatives Find the derivatives of the following functions.
f(x) = ln(3 sin² 4x)

Verified step by step guidance

1. Recognize that the function is a composition of functions: \(f(x) = \ln(3 \sin^{2}(4x))\). We will need to use the chain rule and properties of logarithms to differentiate it.
2. Use the logarithm property to simplify the function before differentiating: \(\ln(3 \sin^{2}(4x)) = \ln(3) + \ln(\sin^{2}(4x)) = \ln(3) + 2 \ln(\sin(4x))\). Since \(\ln(3)\) is a constant, its derivative is zero.
3. Focus on differentiating \$2 \ln(\sin(4x))\(. Apply the constant multiple rule: the derivative is \)2\( times the derivative of \)\ln(\sin(4x))$.
4. Differentiate \(\ln(\sin(4x))\) using the chain rule: the derivative of \(\ln(u)\) is \(\frac{1}{u} \cdot \frac{du}{dx}\). Here, \(u = \sin(4x)\), so \(\frac{du}{dx} = \cos(4x) \cdot 4\) by the chain rule.
5. Combine all parts to write the derivative: \(f'(x) = 2 \cdot \frac{1}{\sin(4x)} \cdot \cos(4x) \cdot 4\). Simplify this expression as much as possible to get the final derivative.

Key Concepts

Chain Rule

The chain rule is used to differentiate composite functions by taking the derivative of the outer function evaluated at the inner function and multiplying it by the derivative of the inner function. For example, if y = f(g(x)), then dy/dx = f'(g(x)) * g'(x). This rule is essential when differentiating functions like ln(3 sin² 4x), which involve nested functions.

Derivative of the Natural Logarithm Function

The derivative of ln(u), where u is a differentiable function of x, is (1/u) * du/dx. This means you first find the derivative of the inside function u and then divide by u itself. This property simplifies differentiating logarithmic functions such as ln(3 sin² 4x).

Derivative of Trigonometric Functions and Power Rule

To differentiate expressions like sin²(4x), you apply the power rule and the chain rule together. The power rule states that d/dx [u^n] = n u^(n-1) du/dx, and the derivative of sin(4x) involves the chain rule: cos(4x) * 4. Combining these helps find the derivative of sin²(4x).