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91Ó°ÊÓ

If \(y=\frac{1}{x}+\frac{2}{x^{2}}+\frac{3}{x^{3}}\), find \(\frac{d y}{d x}\)

Short Answer

Expert verified
-x^{-2}-4*x^{-3}+9*x^{-4}

Step by step solution

01

Rewriting the function y

Rewrite the terms in the given function in the form \(x^n\) to make them amenable to the power rule. So, \(y=x^{-1}+2*x^{-2}+3*x^{-3}\)
02

Apply the Power Rule

Applying the power rule \(d/dx[x^n]=n*x^{n-1}\), the derivative of each term of y with respect to x is calculated as follows: \n\n - \(d/dx[x^{-1}]= -1*x^{-2}\), \n\n - \(d/dx[2*x^{-2}]= -4*x^{-3}\), \n\n - \(d/dx[3*x^{-3}]= 9*x^{-4}\)
03

Simplify the Resultant Derivative

Combine all the derived terms for the complete derivative of y. Therefore, \(\frac{d y}{d x} = -x^{-2}-4*x^{-3}+9*x^{-4}\)

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