Question

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

enhanced feedback
please try again using the combination of chain rule and the formula \\(\frac{d}{dx}(a^{x}) = a^{x}cdot\ln(a)\\). identify (u) such that the function has the form (a^{u}). then use the chain rule.

Explanation:

Step1: Identify the outer - inner functions

Let $u = t^3$ and $y = 5^u$.

Step2: Find the derivative of the outer function

The derivative of $y = a^u$ with respect to $u$ is $\frac{dy}{du}=a^u\ln(a)$. Here $a = 5$, so $\frac{dy}{du}=5^u\ln(5)$.

Step3: Find the derivative of the inner function

The derivative of $u=t^3$ with respect to $t$ is $\frac{du}{dt}=3t^2$.

Step4: Apply the chain - rule

The chain - rule states that $\frac{dy}{dt}=\frac{dy}{du}\cdot\frac{du}{dt}$. Substituting $\frac{dy}{du}=5^u\ln(5)$ and $\frac{du}{dt}=3t^2$ and $u = t^3$ back in, we get $\frac{dy}{dt}=5^{t^3}\ln(5)\cdot3t^2$.

Answer:

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