Derivative of y=e^xcos^4x

1. Apply the product rule:

   $$\frac{d}{d x} f{\left(x \right)} g{\left(x \right)} = f{\left(x \right)} \frac{d}{d x} g{\left(x \right)} + g{\left(x \right)} \frac{d}{d x} f{\left(x \right)}$$

   $f{\left(x \right)} = e^{x}$; to find $\frac{d}{d x} f{\left(x \right)}$:

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

   $g{\left(x \right)} = \cos^{4}{\left(x \right)}$; to find $\frac{d}{d x} g{\left(x \right)}$:

   1. Let $u = \cos{\left(x \right)}$.

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

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

      1. The derivative of cosine is negative sine:

         $$\frac{d}{d x} \cos{\left(x \right)} = - \sin{\left(x \right)}$$

      The result of the chain rule is:

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

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

2. Now simplify:

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

The answer is:

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