Find the Derivative - d/dx 2/x-2/(x^3)+1/(x^4)

Problem

d()/d(x)*(2/x−2/(x3)+1/(x4))

Solution

1. Rewrite the expression using negative exponents to prepare for the power rule.

2/x−2/(x3)+1/(x4)=2*x(−1)−2*x(−3)+x(−4)

2. Apply the sum and difference rules for derivatives to differentiate each term independently.

d()/d(x)*(2*x(−1)−2*x(−3)+x(−4))=d()/d(x)*2*x(−1)−d()/d(x)*2*x(−3)+d()/d(x)*x(−4)

3. Apply the power rule, which states d(xn)/d(x)=n*x(n−1) to each term.

d()/d(x)*2*x(−1)=2*(−1)*x(−2)=−2*x(−2)

d()/d(x)*2*x(−3)=2*(−3)*x(−4)=−6*x(−4)

d()/d(x)*x(−4)=−4*x(−5)

4. Combine the results and simplify the signs.

−2*x(−2)−(−6*x(−4))+(−4*x(−5))=−2*x(−2)+6*x(−4)−4*x(−5)

5. Convert the expression back into fraction form with positive exponents.

−2*x(−2)+6*x(−4)−4*x(−5)=−2/(x2)+6/(x4)−4/(x5)

Final Answer

d()/d(x)*(2/x−2/(x3)+1/(x4))=−2/(x2)+6/(x4)−4/(x5)

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