To find $g'(x)$ in two different ways, we can use the Fundamental Theorem of Calculus:

(a) Using Part One of the Fundamental Theorem of Calculus:

The first part of the Fundamental Theorem of Calculus states that if $g(x) = \int_{a}^{x} f(t) \, dt$, then the derivative is:

$g'(x) = f(x)$

For $g(x) = \int_{1}^{x} t^4 \, dt$, the derivative is:

$g'(x) = x^4$

(b) Evaluating the Integral and Differentiating:

First, compute the integral:

$g(x) = \int_{1}^{x} t^4 \, dt$

This results in:

$g(x) = \left[ \frac{t^5}{5} \right]_{1}^{x} = \frac{x^5}{5} - \frac{1^5}{5} = \frac{x^5}{5} - \frac{1}{5}$

Now, differentiate $g(x)$:

$g'(x) = \frac{d}{dx} \left( \frac{x^5}{5} - \frac{1}{5} \right) = \frac{5x^4}{5} = x^4$

So in both approaches, $g'(x) = x^4$.