Second Derivatives

Find y'' in Exercises 59–64.

y = x(2x + 1)⁴

검증된 단계별 안내

First, identify the function y = x(2x + 1)^4. We need to find the second derivative y''.

To find the first derivative y', apply the product rule. The product rule states that if you have a function y = u(x)v(x), then y' = u'(x)v(x) + u(x)v'(x). Here, let u(x) = x and v(x) = (2x + 1)^4.

Calculate the derivatives: u'(x) = 1 and v'(x) using the chain rule. For v(x) = (2x + 1)^4, let w = 2x + 1, then v(x) = w^4. The derivative v'(x) = 4w^3 * (dw/dx) = 4(2x + 1)^3 * 2.

Substitute the derivatives back into the product rule: y' = 1 * (2x + 1)^4 + x * 8(2x + 1)^3.

Simplify y' and then find the second derivative y'' by differentiating y' again. Use the product rule and chain rule as necessary to find y''.

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주요 개념

질문에 올바르게 답하기 위해 반드시 이해해야 하는 핵심 개념들은 다음과 같습니다.

Product Rule

The product rule is a fundamental differentiation rule used when finding the derivative of a product of two functions. If you have two functions u(x) and v(x), the derivative of their product is given by u'(x)v(x) + u(x)v'(x). This rule is essential for differentiating expressions like y = x(2x + 1)⁴, where x is multiplied by another function.

Chain Rule

The chain rule is used to differentiate composite functions, where one function is nested inside another. If you have a function y = f(g(x)), the derivative is f'(g(x)) * g'(x). In the given problem, the chain rule helps differentiate the term (2x + 1)⁴, treating it as a composite function with an inner function g(x) = 2x + 1.