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

Problem

d()/d(x)*(3/x−2/(x3)+1/(x4))

Solution

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

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

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

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

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

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

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

d(x(−4))/d(x)=−4*x(−5)

4. Combine the results and simplify the signs.

−3*x(−2)−(−6*x(−4))+(−4*x(−5))=−3*x(−2)+6*x(−4)−4*x(−5)

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

−3*x(−2)+6*x(−4)−4*x(−5)=−3/(x2)+6/(x4)−4/(x5)

Final Answer

d()/d(x)*(3/x−2/(x3)+1/(x4))=−3/(x2)+6/(x4)−4/(x5)

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