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 = (2x + 1)⁵

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Step 1: Identify the inner function u = g(x). In this case, the inner function is u = 2x + 1.

Step 2: Express the original function y in terms of u. Here, y = f(u) = u⁵.

Step 3: Differentiate y = f(u) with respect to u to find dy/du. The derivative of u⁵ with respect to u is dy/du = 5u⁴.

Step 4: Differentiate u = g(x) with respect to x to find du/dx. The derivative of 2x + 1 with respect to x is du/dx = 2.

Step 5: Use the chain rule to find dy/dx as a function of x. The chain rule states that dy/dx = (dy/du) * (du/dx). Substitute the expressions found in steps 3 and 4: dy/dx = 5u⁴ * 2. Replace u with 2x + 1 to express dy/dx in terms of x.

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Key Concepts

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

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 this context, we express y as a function of u, where u is a simpler function of x. This allows us to break down complex functions into manageable parts, facilitating differentiation.

Chain Rule

The chain rule is a fundamental principle in calculus used to differentiate composite functions. It states that if a function y depends on u, which in turn depends on x, 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 rule is essential for finding derivatives of nested functions.

Power Rule

The power rule is a basic differentiation technique that states if y = x^n, then dy/dx = n*x^(n-1). This rule simplifies the process of finding derivatives of polynomial functions. In the given problem, applying the power rule to the function y = (2x + 1)⁵ will help in calculating dy/du before using the chain rule to find dy/dx.

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