Calculate the derivative of the following functions. In some cases, it is useful to use the properties of logarithms to simplify the functions before computing f'(x).

y = (cos x) In cos²x

검증된 단계별 안내

First, identify the function y = (cos x) ln(cos²x). Notice that it is a product of two functions: u(x) = cos(x) and v(x) = ln(cos²x).

Apply the product rule for derivatives, which states that if y = u(x) * v(x), then y' = u'(x) * v(x) + u(x) * v'(x).

Calculate the derivative of u(x) = cos(x). The derivative u'(x) is -sin(x).

Simplify v(x) = ln(cos²x) using the properties of logarithms: ln(cos²x) = 2 ln(cos(x)). Now, find the derivative v'(x). The derivative of ln(cos(x)) is -tan(x), so v'(x) = 2(-tan(x)) = -2tan(x).

Substitute u'(x), v(x), u(x), and v'(x) into the product rule formula: y' = (-sin(x)) * ln(cos²x) + cos(x) * (-2tan(x)). Simplify the expression to find the derivative.

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주요 개념

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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, representing the slope of the tangent line to the curve of the function at any given point. The derivative is denoted as f'(x) or dy/dx and can be calculated using various rules, such as the product rule, quotient rule, and chain rule.

Logarithmic Properties

Logarithmic properties are rules that simplify the manipulation of logarithmic expressions. Key properties include the product rule (log(a*b) = log(a) + log(b)), the quotient rule (log(a/b) = log(a) - log(b)), and the power rule (log(a^b) = b*log(a)). These properties are particularly useful in calculus for simplifying complex functions before differentiation, making it easier to compute derivatives.

Chain Rule

The chain rule is a fundamental technique in calculus used to differentiate composite functions. It states that if a function y = f(g(x)) is composed of two functions, 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 dealing with functions that are nested within each other, allowing for the correct application of differentiation.

Calculate the derivative of the following functions. In some cases, it is useful to use the properties of logarithms to simplify the functions before computing f'(x).y = (cos x) In cos²x