Find the Derivative - d/dt f(t)=(cos(t))/t

Problem

d()/d(t)cos(t)/t

Solution

1. Identify the function as a quotient of two functions, u(t)=cos(t) and v(t)=t which requires the quotient rule.

2. Recall the quotient rule formula for a derivative: d()/d(t)u/v=(vd(u)/d(t)−ud(v)/d(t))/(v2)

3. Differentiate the numerator and denominator separately: d(cos(t))/d(t)=−sin(t) and d(t)/d(t)=1

4. Substitute these derivatives into the quotient rule formula: (t*(−sin(t))−cos(t)*(1))/(t2)

5. Simplify the expression by distributing the terms in the numerator.

Final Answer

d()/d(t)cos(t)/t=(−t*sin(t)−cos(t))/(t2)

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