Derivative of (x+1)^2cos5x

1. Apply the product rule:

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

   $f{\left(x \right)} = \left(x + 1\right)^{2}$; to find $\frac{d}{d x} f{\left(x \right)}$:

   1. Let $u = x + 1$.

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

   3. 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:

      $$2 x + 2$$

   $g{\left(x \right)} = \cos{\left(5 x \right)}$; to find $\frac{d}{d x} g{\left(x \right)}$:

   1. Let $u = 5 x$.

   2. The derivative of cosine is negative sine:

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

   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:

      $$- 5 \sin{\left(5 x \right)}$$

   The result is: $- 5 \left(x + 1\right)^{2} \sin{\left(5 x \right)} + \left(2 x + 2\right) \cos{\left(5 x \right)}$

2. Now simplify:

   $$\left(x + 1\right) \left(- 5 \left(x + 1\right) \sin{\left(5 x \right)} + 2 \cos{\left(5 x \right)}\right)$$

The answer is:

$$\left(x + 1\right) \left(- 5 \left(x + 1\right) \sin{\left(5 x \right)} + 2 \cos{\left(5 x \right)}\right)$$