Problem 14 - College Algebra and Calculus: An Applied Approach

Question

Find the derivative of the function. $$ y=x^{3}-9 x^{2}+2 $$

Step-by-Step Solution

Answer

The derivative of the function \(y = x^{3}-9 x^{2}+2\) is \(y' = 3x^{2}-18x\).

Step 1: Identify the Terms

The given function has three terms: one is a cubic term (\(x^3\)), one is a squared term (\(-9x^2\)), and one is a constant term (2). We will find the derivative of each term separately and then combine them to get the derivative of the whole function.

Step 2: Find the Derivative of the Cubic Term

Use the power rule to find the derivative of \(x^3\). According to the power rule, the derivative of \(x^n\) is \(nx^{n-1}\). Here, n=3 so the derivative of \(x^3\) is \(3x^{3-1}\) which simplifies to \(3x^2\).

Step 3: Find the Derivative of the Squared Term

Now, let's find the derivative of \(-9x^2\). Again, using the power rule, the derivative is \(-18x^{2-1}\) which simplifies to \(-18x\).

Step 4: Find the Derivative of the Constant Term

The derivative of a constant term is zero. So, the derivative of 2 is 0.

Step 5: Combine the Derivatives

Finally, combine the derivatives from the three terms to find the derivative of the whole function. The derivative of \(x^3 - 9x^2 + 2\) is \(3x^2 - 18x + 0\) which simplifies to \(3x^2 - 18x\).