Derivative of 3*e^(7*x)+log(3*x)-cos(3*x+1)

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

   1. Differentiate $3 e^{7 x} + \log{\left(3 x \right)}$ term by term:

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

         1. Let $u = 7 x$.

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

         3. Then, apply the chain rule. Multiply by $\frac{d}{d x} 7 x$:

            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: $7$

            The result of the chain rule is:

            $$7 e^{7 x}$$

         So, the result is: $21 e^{7 x}$

      2. Let $u = 3 x$.

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

      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:

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

      The result is: $21 e^{7 x} + \frac{1}{x}$

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

      1. Let $u = 3 x + 1$.

      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 $\frac{d}{d x} \left(3 x + 1\right)$:

         1. Differentiate $3 x + 1$ term by term:

            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: $3$

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

            The result is: $3$

         The result of the chain rule is:

         $$- 3 \sin{\left(3 x + 1 \right)}$$

      So, the result is: $3 \sin{\left(3 x + 1 \right)}$

   The result is: $21 e^{7 x} + 3 \sin{\left(3 x + 1 \right)} + \frac{1}{x}$

2. Now simplify:

   $$21 e^{7 x} + 3 \sin{\left(3 x + 1 \right)} + \frac{1}{x}$$

The answer is:

$$21 e^{7 x} + 3 \sin{\left(3 x + 1 \right)} + \frac{1}{x}$$