Derivative of y=e^(5x)*sin(4x)

1. Apply the product rule:

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

   $f{\left(x \right)} = e^{5 x}$; to find $\frac{d}{d x} f{\left(x \right)}$:

   1. Let $u = 5 x$.

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

   3. Then, apply the chain rule. Multiply by $\frac{d}{d x} 5 x$:

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

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

         So, the result is: $5$

      The result of the chain rule is:

      $$5 e^{5 x}$$

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

   1. Let $u = 4 x$.

   2. The derivative of sine is cosine:

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

   3. Then, apply the chain rule. Multiply by $\frac{d}{d x} 4 x$:

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

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

         So, the result is: $4$

      The result of the chain rule is:

      $$4 \cos{\left(4 x \right)}$$

   The result is: $5 e^{5 x} \sin{\left(4 x \right)} + 4 e^{5 x} \cos{\left(4 x \right)}$

2. Now simplify:

   $$\left(5 \sin{\left(4 x \right)} + 4 \cos{\left(4 x \right)}\right) e^{5 x}$$

The answer is:

$$\left(5 \sin{\left(4 x \right)} + 4 \cos{\left(4 x \right)}\right) e^{5 x}$$