Find the Derivative - d/dx (cos(x))/(x^3)

Problem

d()/d(x)cos(x)/(x^3)

Solution

1. Identify the rule needed for differentiation. Since the expression is a quotient of two functions, apply the quotient rule: d()/d(x)u/v=(vd(u)/d(x)−ud(v)/d(x))/(v^2)

2. Assign the variables for the numerator and denominator. Let u=cos(x) and v=x^3

3. Differentiate the individual components. The derivative of the numerator is d(cos(x))/d(x)=−sin(x) and the derivative of the denominator is d(x^3)/d(x)=3*x^2

4. Substitute these values into the quotient rule formula.

d()/d(x)cos(x)/(x^3)=(x^3*(−sin(x))−cos(x)*(3*x^2))/((x^3)^2)

5. Simplify the expression by multiplying terms and simplifying the denominator.

d()/d(x)cos(x)/(x^3)=(−x^3*sin(x)−3*x^2*cos(x))/(x^6)

6. Factor out the common term x^2 from the numerator to reduce the fraction.

d()/d(x)cos(x)/(x^3)=(x^2*(−x*sin(x)−3*cos(x)))/(x^6)

7. Divide the numerator and denominator by x^2

d()/d(x)cos(x)/(x^3)=(−x*sin(x)−3*cos(x))/(x^4)

Final Answer

d()/d(x)cos(x)/(x^3)=(−x*sin(x)−3*cos(x))/(x^4)

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