Find the Derivative - d/dx 4 square root of x-5/(x^2)

Problem

d()/d(x)*(4√(,x)−5/(x2))

Solution

1. Rewrite the expression using power notation to make differentiation easier.

√(,x)=x(1/2)

5/(x2)=5*x(−2)

2. Apply the sum and difference rule for derivatives.

d()/d(x)*(4*x(1/2)−5*x(−2))=(d(4)*x(1/2))/d(x)−(d(5)*x(−2))/d(x)

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

(d(4)*x(1/2))/d(x)=4⋅1/2*x(−1/2)=2*x(−1/2)

(d(5)*x(−2))/d(x)=5⋅(−2)*x(−3)=−10*x(−3)

4. Combine the results and simplify the signs.

2*x(−1/2)−(−10*x(−3))=2*x(−1/2)+10*x(−3)

5. Convert the expression back into radical and fraction form.

2*x(−1/2)=2/√(,x)

10*x(−3)=10/(x3)

Final Answer

d()/d(x)*(4√(,x)−5/(x2))=2/√(,x)+10/(x3)

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