find the derivative of the function.\n\n$f(t)=5^{t^{3}}$\n\n$f(t)=5^{t^{3}}3t^{2}$

Explanation:

Step1: Use the chain rule

The chain rule states that if (y = a^{u}), then (y^\prime=a^{u}\ln a\cdot u^\prime). Here (a = 5), (u=t^{3}), and (u^\prime = 3t^{2}).

Step2: Substitute into the chain - rule formula

(f(t)=5^{t^{3}}), so (f^\prime(t)=5^{t^{3}}\ln 5\cdot\frac{d}{dt}(t^{3})).

Step3: Differentiate (t^{3})

Since (\frac{d}{dt}(t^{n})=nt^{n - 1}), for (n = 3), (\frac{d}{dt}(t^{3})=3t^{2}).

Step4: Write the final derivative

(f^\prime(t)=5^{t^{3}}\ln 5\cdot3t^{2}=3t^{2}\ln 5\cdot5^{t^{3}})

Answer:

(3t^{2}\ln 5\cdot5^{t^{3}})