Derivative of exp(x)+(x-1)^4

1. Differentiate $\left(x - 1\right)^{4} + e^{x}$ term by term:

   1. The derivative of $e^{x}$ is itself.

   2. Let $u = x - 1$.

   3. Apply the power rule: $u^{4}$ goes to $4 u^{3}$

   4. Then, apply the chain rule. Multiply by $\frac{d}{d x} \left(x - 1\right)$:

      1. Differentiate $x - 1$ term by term:

         1. Apply the power rule: $x$ goes to $1$

         2. The derivative of the constant $-1$ is zero.

         The result is: $1$

      The result of the chain rule is:

      $$4 \left(x - 1\right)^{3}$$

   The result is: $4 \left(x - 1\right)^{3} + e^{x}$

2. Now simplify:

   $$4 \left(x - 1\right)^{3} + e^{x}$$

The answer is:

$$4 \left(x - 1\right)^{3} + e^{x}$$