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Chapter 7: Problem 14

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

Short Answer

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

Step by step solution

01

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.
02

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 \(3x^2\).
03

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 \(-18x\).
04

Find the Derivative of the Constant Term

The derivative of a constant term is zero. So, the derivative of 2 is 0.
05

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 \(3x^2 - 18x\).

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