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

Problem

d()/d(t)t/((t−1)2)

Solution

1. Identify the rule needed for differentiation. Since the expression is a quotient of two functions, apply the quotient rule: d()/d(t)u/v=(vd(u)/d(t)−ud(v)/d(t))/(v2)

2. Assign the variables for the numerator and denominator. Let u=t and v=(t−1)2

3. Differentiate the individual components. The derivative of the numerator is d(t)/d(t)=1 For the denominator, apply the chain rule: d(t−1)/d(t)=2*(t−1)⋅1=2*(t−1)

4. Substitute these values into the quotient rule formula.

d()/d(t)t/((t−1)2)=((t−1)2*(1)−t*(2*(t−1)))/(((t−1)2)2)

5. Factor out the common term (t−1) from the numerator to simplify the expression.

d()/d(t)t/((t−1)2)=((t−1)*((t−1)−2*t))/((t−1)4)

6. Simplify the fraction by canceling one (t−1) term and combining like terms in the numerator.

d()/d(t)t/((t−1)2)=(t−1−2*t)/((t−1)3)

d()/d(t)t/((t−1)2)=(−t−1)/((t−1)3)

Final Answer

d()/d(t)t/((t−1)2)=−(t+1)/((t−1)3)

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