7–28. Derivatives Evaluate the following derivatives.

d/dx ((ln 2x)⁻⁵)

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Step 1: Recognize that the given function is ((ln(2x))⁻⁵). This is a composite function, so we will need to use the chain rule to differentiate it.

Step 2: Apply the chain rule. Let u = ln(2x), so the function becomes u⁻⁵. The derivative of u⁻⁵ with respect to u is -5u⁻⁶.

Step 3: Differentiate u = ln(2x) with respect to x. Using the chain rule again, the derivative of ln(2x) is (1/(2x)) * 2, which simplifies to 1/x.

Step 4: Combine the results from Step 2 and Step 3. Multiply the derivative of u⁻⁵ (-5u⁻⁶) by the derivative of u (1/x). Substitute u = ln(2x) back into the expression.

Step 5: The final derivative is -5(ln(2x))⁻⁶ * (1/x). This is the simplified form of the derivative.

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Derivatives

A derivative represents the rate of change of a function with respect to a variable. It is a fundamental concept in calculus that allows us to determine how a function behaves as its input changes. The notation d/dx indicates differentiation with respect to the variable x, and derivatives can be computed 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 a composite function. If a function y is defined as a function of u, which in turn is a function of x (y = f(u) and u = g(x)), the chain rule states that the derivative dy/dx is the product of the derivative of f with respect to u and the derivative of g with respect to x. This is essential for differentiating functions like (ln(2x))⁻⁵, where the inner function is ln(2x).

Natural Logarithm

The natural logarithm, denoted as ln, is the logarithm to the base e, where e is approximately 2.71828. It is an important function in calculus, particularly in differentiation and integration. The derivative of ln(x) is 1/x, and when dealing with expressions involving ln, understanding its properties and how to differentiate it is crucial for solving problems involving logarithmic functions.

7–28. Derivatives Evaluate the following derivatives.d/dx ((ln 2x)⁻⁵)