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

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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.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

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.

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