In Exercises 9–18, write the function in the form y = f(u) and u = g(x). Then find dy/dx as a function of x.

y = (4 − 3x)⁹

검증된 단계별 안내

Step 1: Identify the inner function u = g(x). In this case, the expression inside the parentheses is the inner function, so we have u = 4 - 3x.

Step 2: Express the original function y in terms of u. Since y = (4 - 3x)⁹, we can rewrite it as y = f(u) = u⁹.

Step 3: Differentiate y = f(u) with respect to u. Using the power rule, the derivative of u⁹ with respect to u is dy/du = 9u⁸.

Step 4: Differentiate u = g(x) with respect to x. The derivative of u = 4 - 3x with respect to x is du/dx = -3.

Step 5: Apply the chain rule to find dy/dx. The chain rule states that dy/dx = (dy/du) * (du/dx). Substitute the derivatives found in steps 3 and 4: dy/dx = 9u⁸ * (-3). Finally, substitute u = 4 - 3x back into the expression to get dy/dx as a function of x.

비슷한 문제에 대한 검증된 영상 답변:

이 영상 해법은 위 문제에 도움이 된다고 튜터들이 추천한 것입니다.

영상 길이:

도움이 되었나요?

주요 개념

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

Chain Rule

The Chain Rule is a fundamental theorem in calculus used to differentiate composite functions. It states that if a function y is defined as a function of u, which in turn is a function of x, then the derivative dy/dx can be found by multiplying the derivative of y with respect to u (dy/du) by the derivative of u with respect to x (du/dx). This allows for the differentiation of complex functions by breaking them down into simpler parts.

Function Composition

Function composition involves creating a new function by combining two functions, where the output of one function becomes the input of another. In the context of the given problem, we express y as a function of u, and u as a function of x. Understanding how to manipulate and express these relationships is crucial for applying the Chain Rule effectively and finding derivatives.

Implicit Differentiation

Implicit differentiation is a technique used to differentiate equations where y is not explicitly solved for x. In cases where y is defined in terms of another variable u, and u in terms of x, implicit differentiation allows us to find dy/dx without isolating y. This method is particularly useful when dealing with complex functions or when the relationship between variables is not straightforward.

In Exercises 9–18, write the function in the form y = f(u) and u = g(x). Then find dy/dx as a function of x.y = (4 − 3x)⁹