Find ds/dt s=t^2 natural log of |t|

Problem

d()/d(t)*t2*ln(t)

Solution

1. Identify the rule needed for differentiation. Since the expression is a product of two functions, u=t2 and v=ln(t) use the product rule: d(s())/d(t)=ud(v)/d(t)+vd(u)/d(t)

2. Differentiate the first part of the product. The derivative of t2 with respect to t is 2*t

3. Differentiate the second part of the product. The derivative of ln(t) with respect to t is 1/t for all t≠0

4. Apply the product rule formula by substituting the functions and their derivatives.

d(s())/d(t)=t2d(ln(t))/d(t)+ln(t)d(t2)/d(t)

5. Substitute the calculated derivatives into the expression.

d(s())/d(t)=t21/t+ln(t)*(2*t)

6. Simplify the resulting expression by canceling t in the first term and rearranging the second term.

d(s())/d(t)=t+2*t*ln(t)

7. Factor out the common term t to reach the final simplified form.

d(s())/d(t)=t*(1+2*ln(t))

Final Answer

d(s())/d(t)=t+2*t*ln(t)

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