Derivative of e^(2t)cost+e^(2t)sint

1. Differentiate $e^{2 t} \sin{\left(t \right)} + e^{2 t} \cos{\left(t \right)}$ term by term:

   1. Apply the product rule:

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

      $f{\left(t \right)} = e^{2 t}$; to find $\frac{d}{d t} f{\left(t \right)}$:

      1. Let $u = 2 t$.

      2. The derivative of $e^{u}$ is itself.

      3. Then, apply the chain rule. Multiply by $\frac{d}{d t} 2 t$:

         1. The derivative of a constant times a function is the constant times the derivative of the function.

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

            So, the result is: $2$

         The result of the chain rule is:

         $$2 e^{2 t}$$

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

      1. The derivative of cosine is negative sine:

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

      The result is: $- e^{2 t} \sin{\left(t \right)} + 2 e^{2 t} \cos{\left(t \right)}$

   2. Apply the product rule:

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

      $f{\left(t \right)} = e^{2 t}$; to find $\frac{d}{d t} f{\left(t \right)}$:

      1. Let $u = 2 t$.

      2. The derivative of $e^{u}$ is itself.

      3. Then, apply the chain rule. Multiply by $\frac{d}{d t} 2 t$:

         1. The derivative of a constant times a function is the constant times the derivative of the function.

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

            So, the result is: $2$

         The result of the chain rule is:

         $$2 e^{2 t}$$

      $g{\left(t \right)} = \sin{\left(t \right)}$; to find $\frac{d}{d t} g{\left(t \right)}$:

      1. The derivative of sine is cosine:

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

      The result is: $2 e^{2 t} \sin{\left(t \right)} + e^{2 t} \cos{\left(t \right)}$

   The result is: $e^{2 t} \sin{\left(t \right)} + 3 e^{2 t} \cos{\left(t \right)}$

2. Now simplify:

   $$\left(\sin{\left(t \right)} + 3 \cos{\left(t \right)}\right) e^{2 t}$$

The answer is:

$$\left(\sin{\left(t \right)} + 3 \cos{\left(t \right)}\right) e^{2 t}$$