Find the Derivative - d/dt (e^(-t))/(1+t^2)

Problem

d()/d(t)(e^(−t))/(1+t^2)

Solution

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

2. Assign the numerator and denominator functions. Let u=e^(−t) and v=1+t^2

3. Differentiate the individual components. Using the chain rule for u we find d(e^(−t))/d(t)=−e^(−t) For v we find d(1+t^2)/d(t)=2*t

4. Substitute these components into the quotient rule formula.

d()/d(t)(e^(−t))/(1+t^2)=((1+t^2)*(−e^(−t))−(e^(−t))*(2*t))/((1+t^2)^2)

5. Factor out the common term e^(−t) from the numerator to simplify the expression.

d()/d(t)(e^(−t))/(1+t^2)=(−e^(−t)*(1+t^2)−2*t*e^(−t))/((1+t^2)^2)

d()/d(t)(e^(−t))/(1+t^2)=(−e^(−t)*(1+t^2+2*t))/((1+t^2)^2)

6. Recognize the quadratic pattern in the numerator. The term t^2+2*t+1 is a perfect square trinomial, (t+1)^2

d()/d(t)(e^(−t))/(1+t^2)=(−e^(−t)*(t+1)^2)/((1+t^2)^2)

Final Answer

d()/d(t)(e^(−t))/(1+t^2)=−(e^(−t)*(t+1)^2)/((1+t^2)^2)

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