Problem 21 - College Algebra and Calculus: An Applied Approach

Question

Find the derivative of the function. $$ y=4 x^{-2}+2 x^{2} $$

Step-by-Step Solution

Answer

The derivative of the function \(y=4x^{-2}+2x^{2}\) is \(-8x^{-3} + 4x\).

Step 1: Identify the Terms of the Function

The given function is \(y=4x^{-2}+2x^{2}\). Here, there are two terms, \(4x^{-2}\) and \(2x^{2}\). From the power rule of differentiation, the derivative of \(x^n\), where n is any real number, is \(nx^{n-1}\). Therefore, for each term, the derivative can be found by multiplying the entire term by the power of x, and then subtracting one from the power.

Step 2: Find the Derivative of the First term

The first term is \(4x^{-2}\). When differentiating, the derivative becomes \(-2*4*x^{-2-1}\) which simplifies to \(-8x^{-3}\).

Step 3: Find the Derivative of the Second Term

The second term is \(2x^{2}\). When differentiating, the derivative becomes \(2*2*x^{2-1}\) which simplifies to \(4x\).

Step 4: Combine the Derivatives of the Terms

Finally, to find the derivative of the entire function, the derivatives of the individual terms are combined. Therefore, the derivative of the given function, \(y=4x^{-2} + 2x^{2}\), is \(-8x^{-3} + 4x\).