Find the Derivative - d/dt y = natural log of t^5

Problem

d(ln(t5))/d(t)

Solution

1. Identify the function as a composition of the natural logarithm and a power function, requiring the use of the chain rule.

2. Apply the chain rule for the natural logarithm, which states that d(ln(u))/d(t)=1/u⋅d(u)/d(t)

3. Substitute u=t5 into the chain rule formula.

4. Differentiate the inner function t5 using the power rule to get 5*t4

5. Multiply the results together:

1/(t5)⋅5*t4

6. Simplify the expression by canceling the common factors of t

(5*t4)/(t5)=5/t

Final Answer

d(ln(t5))/d(t)=5/t

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