Find the Derivative - d/dt g(t)=(t- square root of t)/(t^(1/3))

Problem

d()/d(t)*g(t)=(t−√(,t))/(t(1/3))

Solution

1. Rewrite the expression by expressing the square root as a power and splitting the fraction into two separate terms.

g(t)=(t−t(1/2))/(t(1/3))

g(t)=t/(t(1/3))−(t(1/2))/(t(1/3))

2. Simplify the terms using the laws of exponents, specifically ta/tb=t(a−b)

g(t)=t(1−1/3)−t(1/2−1/3)

g(t)=t(2/3)−t(1/6)

3. Apply the power rule for derivatives, which states that d(tn)/d(t)=n*t(n−1) to each term.

d(g(t))/d(t)=2/3*t(2/3−1)−1/6*t(1/6−1)

d(g(t))/d(t)=2/3*t(−1/3)−1/6*t(−5/6)

4. Rewrite the result using positive exponents or radical notation if desired.

d(g(t))/d(t)=2/(3*t(1/3))−1/(6*t(5/6))

Final Answer

d()/d(t)(t−√(,t))/(t(1/3))=2/(3*t(1/3))−1/(6*t(5/6))

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