Find the Derivative - d/dx 1/x-3sin(x)

Problem

d()/d(x)*(1/x−3*sin(x))

Solution

1. Apply the sum rule for differentiation to separate the terms.

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

2. Rewrite the first term using a negative exponent to prepare for the power rule.

1/x=x(−1)

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

d(x(−1))/d(x)=−1*x(−2)

4. Apply the constant multiple rule and the derivative of the sine function to the second term, where d(sin(x))/d(x)=cos(x)

(d(3)*sin(x))/d(x)=3*cos(x)

5. Simplify the expression by converting the negative exponent back into a fraction.

−1*x(−2)=−1/(x2)

Final Answer

d()/d(x)*(1/x−3*sin(x))=−1/(x2)−3*cos(x)

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