Derivative of 7sqrtx*log5x+cos3x+1

1. Differentiate $\left(7 \sqrt{x} \log{\left(5 x \right)} + \cos{\left(3 x \right)}\right) + 1$ term by term:

   1. Differentiate $7 \sqrt{x} \log{\left(5 x \right)} + \cos{\left(3 x \right)}$ term by term:

      1. Apply the product rule:

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

         $f{\left(x \right)} = 7 \sqrt{x}$; to find $\frac{d}{d x} f{\left(x \right)}$:

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

            1. Apply the power rule: $\sqrt{x}$ goes to $\frac{1}{2 \sqrt{x}}$

            So, the result is: $\frac{7}{2 \sqrt{x}}$

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

         1. Let $u = 5 x$.

         2. The derivative of $\log{\left(u \right)}$ is $\frac{1}{u}$.

         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:

            $$\frac{1}{x}$$

         The result is: $\frac{7 \log{\left(5 x \right)}}{2 \sqrt{x}} + \frac{7}{\sqrt{x}}$

      2. Let $u = 3 x$.

      3. The derivative of cosine is negative sine:

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

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

         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: $3$

         The result of the chain rule is:

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

      The result is: $- 3 \sin{\left(3 x \right)} + \frac{7 \log{\left(5 x \right)}}{2 \sqrt{x}} + \frac{7}{\sqrt{x}}$

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

   The result is: $- 3 \sin{\left(3 x \right)} + \frac{7 \log{\left(5 x \right)}}{2 \sqrt{x}} + \frac{7}{\sqrt{x}}$

2. Now simplify:

   $$\frac{- 6 \sqrt{x} \sin{\left(3 x \right)} + 7 \log{\left(5 x \right)} + 14}{2 \sqrt{x}}$$

The answer is:

$$\frac{- 6 \sqrt{x} \sin{\left(3 x \right)} + 7 \log{\left(5 x \right)} + 14}{2 \sqrt{x}}$$