7–28. Derivatives Evaluate the following derivatives.

d/dx (ln (cos² x))

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Step 1: Recognize that the derivative involves the natural logarithm function ln and the cosine squared function cos²(x). Use the chain rule and logarithmic differentiation to simplify the process.

Step 2: Apply the logarithmic property ln(a^b) = b * ln(a) to rewrite ln(cos²(x)) as 2 * ln(cos(x)). This simplifies the expression and makes differentiation easier.

Step 3: Differentiate the simplified expression 2 * ln(cos(x)) with respect to x. The constant 2 remains unchanged, and you focus on differentiating ln(cos(x)).

Step 4: Use the chain rule to differentiate ln(cos(x)). The derivative of ln(u) is 1/u, and the derivative of cos(x) is -sin(x). Combine these to get (1/cos(x)) * (-sin(x)).

Step 5: Multiply the result from Step 4 by the constant 2 from Step 3. The final derivative is -2 * (sin(x)/cos(x)), which can also be expressed as -2 * tan(x).

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Derivatives

A derivative represents the rate of change of a function with respect to a variable. In calculus, it is a fundamental concept that allows us to determine how a function behaves at any given point. The derivative is often denoted as f'(x) or dy/dx, and it can be interpreted as the slope of the tangent line to the curve of the function at a specific point.

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 g(x) that is composed with another function f(u), where u = g(x), then the derivative of f with respect to x can be found by multiplying the derivative of f with respect to u by the derivative of g with respect to x. This is essential for differentiating complex functions like ln(cos² x).

Logarithmic Differentiation

Logarithmic differentiation is a technique used to differentiate functions that are products or quotients of variables, especially when they involve exponentials or logarithms. By taking the natural logarithm of both sides of an equation, we can simplify the differentiation process. This method is particularly useful for functions like ln(cos² x), as it allows us to apply the properties of logarithms to simplify the expression before differentiating.

7–28. Derivatives Evaluate the following derivatives.d/dx (ln (cos² x))