Problem 23 - College Algebra and Calculus: An Applied Approach

Question

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

Step-by-Step Solution

Answer

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

Step 1 - Identify u and n

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

Step 2 - 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 \(3(2x-7)^2\).

Step 3 - 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.

Step 4 - 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: \(6(2x-7)^2\).