Find the derivatives of the functions in Exercises 1–42.

𝔂 = 3 .
(5x² + sin 2x)³/²

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Identify the function structure: The function is a composition of functions, specifically a power function applied to a sum of a polynomial and a trigonometric function. This suggests the use of the chain rule for differentiation.

Apply the chain rule: The chain rule states that if you have a composite function y = f(g(x)), then the derivative y' = f'(g(x)) * g'(x). Here, let u = 5x² + sin(2x), so y = 3u^(3/2).

Differentiate the outer function: The derivative of 3u^(3/2) with respect to u is (3 * (3/2) * u^(1/2)) = (9/2)u^(1/2).

Differentiate the inner function: Now, find the derivative of u = 5x² + sin(2x). The derivative of 5x² is 10x, and the derivative of sin(2x) is 2cos(2x) (using the chain rule for sin(2x)). Therefore, u' = 10x + 2cos(2x).

Combine the derivatives: Multiply the derivative of the outer function by the derivative of the inner function to get the final derivative: y' = (9/2)(5x² + sin(2x))^(1/2) * (10x + 2cos(2x)).

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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 at any given point. Derivatives can be computed using various rules, such as the power rule, product rule, and chain rule, which simplify the process of finding the rate of change.

Chain Rule

The chain rule is a formula for computing the derivative of a composite function. If a function is composed of two or more functions, the chain rule states that the derivative of the outer function is multiplied by the derivative of the inner function. This is particularly useful when dealing with functions raised to a power, as seen in the given problem, where the outer function is a power and the inner function is a polynomial plus a trigonometric function.

Power Rule

The power rule is a basic rule for finding the derivative of a function of the form f(x) = x^n, where n is a real number. According to this rule, the derivative f'(x) is given by n*x^(n-1). This rule simplifies the differentiation process, especially for polynomial functions, and is often used in conjunction with the chain rule when dealing with more complex expressions.

Find the derivatives of the functions in Exercises 1–42.𝔂 = 3 .(5x² + sin 2x)³/²