Find the Derivative - d/dt cube root of t^2+2 square root of t^3

Problem

d()/d(t)*(√(3,t2)+2√(,t3))

Solution

1. Rewrite the radical expressions as power functions using the rule √(n,tm)=t(m/n)

√(3,t2)=t(2/3)

2√(,t3)=2*t(3/2)

2. Apply the sum rule for derivatives, which allows for differentiating each term separately.

d()/d(t)*(t(2/3)+2*t(3/2))=d(t(2/3))/d(t)+(d(2)*t(3/2))/d(t)

3. Use the power rule d(tn)/d(t)=n*t(n−1) for the first term.

d(t(2/3))/d(t)=2/3*t(2/3−1)

d(t(2/3))/d(t)=2/3*t(−1/3)

4. Use the power rule and the constant multiple rule for the second term.

(d(2)*t(3/2))/d(t)=2⋅3/2*t(3/2−1)

(d(2)*t(3/2))/d(t)=3*t(1/2)

5. Combine the results and rewrite using radical notation if desired.

2/3*t(−1/3)+3*t(1/2)=2/(3√(3,t))+3√(,t)

Final Answer

d()/d(t)*(√(3,t2)+2√(,t3))=2/(3√(3,t))+3√(,t)

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