Find the Derivative - d/dt y=(tsin(t))/(1+t)

Problem

d()/d(t)(t*sin(t))/(1+t)

Solution

1. Identify the rule needed for the derivative of a quotient, which is the quotient rule: d()/d(t)u/v=(vd(u)/d(t)−ud(v)/d(t))/(v2)

2. Define the numerator as u=t*sin(t) and the denominator as v=1+t

3. Differentiate the numerator u using the product rule: d(u)/d(t)=sin(t)+t*cos(t)

4. Differentiate the denominator v d(v)/d(t)=1

5. Substitute these components into the quotient rule formula:

d(y)/d(t)=((1+t)*(sin(t)+t*cos(t))−(t*sin(t))*(1))/((1+t)2)

6. Expand the terms in the numerator:

d(y)/d(t)=(sin(t)+t*cos(t)+t*sin(t)+t2*cos(t)−t*sin(t))/((1+t)2)

7. Simplify the numerator by canceling the t*sin(t) and −t*sin(t) terms:

d(y)/d(t)=(sin(t)+t*cos(t)+t2*cos(t))/((1+t)2)

8. Factor the common t*cos(t) and t2*cos(t) terms if desired:

d(y)/d(t)=(sin(t)+(t+t2)*cos(t))/((1+t)2)

Final Answer

d()/d(t)(t*sin(t))/(1+t)=(sin(t)+t*(1+t)*cos(t))/((1+t)2)

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