Find the derivative of the following functions.
y = In(cos² x)

Verified step by step guidance

First, recognize that the function y = ln(cos²(x)) is a composition of functions. It involves the natural logarithm function ln(u) where u = cos²(x).

Apply the chain rule for differentiation, which states that if y = ln(u), then dy/dx = (1/u) * du/dx. Here, u = cos²(x).

Differentiate u = cos²(x) with respect to x. Use the chain rule again: if u = (cos(x))², then du/dx = 2 * cos(x) * (-sin(x)), because the derivative of cos(x) is -sin(x).

Substitute du/dx into the chain rule formula: dy/dx = (1/cos²(x)) * (2 * cos(x) * (-sin(x))).

Simplify the expression: dy/dx = -2 * sin(x) / cos(x), which can be further simplified to dy/dx = -2 * tan(x), using the identity tan(x) = sin(x)/cos(x).

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Derivative

The derivative of a function measures how the function's output value changes as its input value changes. It is a fundamental concept in calculus that provides the slope of the tangent line to the curve of the function at any given point. The derivative is often denoted as f'(x) or dy/dx and can be calculated using various rules, such as the power rule, product rule, and chain rule.

Chain Rule

The chain rule is a formula for computing the derivative of the composition of two or more functions. It states that if you have a function y = f(g(x)), the derivative can be found by multiplying the derivative of the outer function f with the derivative of the inner function g. This rule is essential when differentiating functions that are nested within each other, such as logarithmic and trigonometric functions.

Logarithmic Differentiation

Logarithmic differentiation is a technique used to differentiate functions that are products or quotients of variables raised to powers. By taking the natural logarithm of both sides of the equation, it simplifies the differentiation process, especially when dealing with complex functions. This method is particularly useful for functions involving exponential growth or decay, as well as for simplifying the differentiation of products of functions.

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