Find the Derivative - d/dt (3 natural log of t)/(at-bt^2)

Problem

d()/d(t)(3*ln(t))/(a*t−b*t2)

Solution

1. Identify the function as a quotient of two functions, u=3*ln(t) and v=a*t−b*t2 which requires the quotient rule.

2. Apply the quotient rule formula, which states that d()/d(t)u/v=(vd(u)/d(t)−ud(v)/d(t))/(v2)

3. Differentiate the numerator u=3*ln(t) with respect to t to get d(u)/d(t)=3/t

4. Differentiate the denominator v=a*t−b*t2 with respect to t to get d(v)/d(t)=a−2*b*t

5. Substitute these derivatives into the quotient rule formula.

d()/d(t)(3*ln(t))/(a*t−b*t2)=((a*t−b*t2)*(3/t)−(3*ln(t))*(a−2*b*t))/((a*t−b*t2)2)

6. Simplify the first term in the numerator by distributing 3/t into (a*t−b*t2)

(a*t−b*t2)*(3/t)=3*a−3*b*t

7. Combine the terms in the numerator to reach the final simplified form.

(3*a−3*b*t−3*ln(t)*(a−2*b*t))/((a*t−b*t2)2)

Final Answer

d()/d(t)(3*ln(t))/(a*t−b*t2)=(3*a−3*b*t−3*(a−2*b*t)*ln(t))/((a*t−b*t2)2)

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