Find the Derivative - d/dx x/4-4/x

Problem

d()/d(x)*(x/4−4/x)

Solution

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

x/4−4/x=1/4*x−4*x(−1)

2. Apply the sum rule for derivatives, which allows for differentiating each term independently.

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

3. Apply the power rule d(xn)/d(x)=n*x(n−1) to both terms.

(d(1/4)*x)/d(x)=1/4

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

4. Combine the results and simplify the signs.

1/4−(−4*x(−2))=1/4+4*x(−2)

5. Convert the negative exponent back into a fraction.

1/4+4/(x2)

Final Answer

d()/d(x)*(x/4−4/x)=1/4+4/(x2)

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