Find the Derivative - d/dt ccos(t)+t^2sin(t)

Problem

d()/d(t)*(c*cos(t)+t2*sin(t))

Solution

1. Apply the sum rule to differentiate each term of the expression separately.

d()/d(t)*(c*cos(t)+t2*sin(t))=(d(c)*cos(t))/d(t)+(d(t2)*sin(t))/d(t)

2. Differentiate the first term by treating c as a constant and applying the rule d(cos(t))/d(t)=−sin(t)

(d(c)*cos(t))/d(t)=−c*sin(t)

3. Apply the product rule to the second term t2*sin(t) using the formula (d(u)*v)/d(t)=ud(v)/d(t)+vd(u)/d(t)

(d(t2)*sin(t))/d(t)=t2d(sin(t))/d(t)+sin(t)d(t2)/d(t)

4. Evaluate the derivatives within the product rule expression.

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

5. Combine the results from the sum rule to find the final derivative.

−c*sin(t)+t2*cos(t)+2*t*sin(t)

6. Factor out the common sin(t) term to simplify the expression.

(2*t−c)*sin(t)+t2*cos(t)

Final Answer

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

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