Problem 16 - College Algebra and Calculus: An Applied Approach

Question

Find the derivative of the function. $$ y=2 x^{3}-x^{2}+3 x-1 $$

Step-by-Step Solution

Answer

\[ y' = 6x^2 - 2x + 3 \]

Step 1: Apply the Power Rule to the First Term

The power rule states that the derivative of \(x^n\) is \(nx^{n-1}\). In the first term, the power of \(x\) is 3. So, applying the power rule gives the derivative as \(3 * 2x^{3-1} = 6x^2\).

Step 2: Apply the Power Rule to the Second Term

In the second term, the power of \(x\) is 2. Applying the power rule gives the derivative as \(2 * 1x^{2-1} = 2x\). As this term is subtracted in the original function, it will also be subtracted in the derivative.

Step 3: Apply the Power Rule to the Third Term

In the third term, the power of \(x\) is 1. Applying the power rule gives the derivative as \(1 * 3x^{1-1} = 3\).

Step 4: Derive the Constant

The derivative of a constant term is simply 0. So, the derivative of -1 is 0.

Step 5: Combine the Three Terms

Add the derived terms together to form the derivative of the complete function. Subtract the second term because it was subtracted in the original function. The derivative is \(6x^2 - 2x + 3\).