Derivative of 2*t*sin(4*x)+1/x

1. Differentiate $2 t \sin{\left(4 x \right)} + \frac{1}{x}$ term by term:

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

      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:

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

      So, the result is: $8 t \cos{\left(4 x \right)}$

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

   The result is: $8 t \cos{\left(4 x \right)} - \frac{1}{x^{2}}$

The answer is:

$$8 t \cos{\left(4 x \right)} - \frac{1}{x^{2}}$$