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Chapter 7: Problem 23

Use the General Power Rule to find the derivative of the function. $$ y=(2 x-7)^{3} $$

Short Answer

Expert verified
The derivative of the function \(y=(2x-7)^3\) is \(y'=6(2x-7)^2\).

Step by step solution

01

- Identify u and n

First, identify u and n from the given problem. Here, \(u=2x-7\) and \(n=3\).
02

- Apply the Power Rule

Apply the Power Rule, which states that the derivative of \(u^n\) is \(n(u^{n-1})u'\). Here, \(n=3\) and \(u=2x-7\). Thus the expression becomes \(3(2x-7)^{3-1}\). This simplifies to \(3(2x-7)^2\).
03

- Find the derivative of u

Now, find the derivative of u, which is the derivative of \(2x-7\). The derivative of \(2x-7\), denoted \(u'\), is 2.
04

- Multiply by u' to get final derivative

Lastly, multiply the expression arrived at Step 2, \(3(2x-7)^2\), by \(u'\), which is 2. This results in the final derivative: \(6(2x-7)^2\).

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