Fresh day Galaxy, let's solve. Step 1: Identify the given function. The function $f(x)$ is defined as an integral: $$f(x)=\int_{a}^{x} t^{3} e^{t} dt$$ Step 2: Apply the Fundamental Theorem of Calculus, Part 1. The Fundamental Theorem of Calculus, Part 1 states that if $F(x) = \int_{a}^{x} g(t) dt$, then $F'(x) = g(x)$. In this case, $g(t) = t^3 e^t$. Therefore, to find $\frac{d}{dx} f(x)$, we substitute $x$ for $t$ in the integrand. Step 3: Differentiate $f(x)$ with respect to $x$. $$\frac{d}{dx} f(x) = \frac{d}{dx} \left( \int_{a}^{x} t^{3} e^{t} dt \right)$$ Applying the theorem, we get: $$\frac{d}{dx} f(x) = x^{3} e^{x}$$ Step 4: Compare the result with the given options. The calculated value is $x^3 e^x$. Option (a) is $e^x (x^3 + 3x^2)$. Option (b) is $x^3 e^x$. Option (c) is $a^3 e^a$. Option (d) is इनमें से कोई नहीं (None of these). The result matches option (b). The final answer is $\boxed{\text{(b)} x^{3} e^{x}}$. Send me the next one 📸