Derivative of y=2x^5+4x+sin(7x)

1. Differentiate $2 x^{5} + 4 x + \sin{\left(7 x \right)}$ 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: $x^{5}$ goes to $5 x^{4}$

      So, the result is: $10 x^{4}$

   2. 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: $4$

   3. Let $u = 7 x$.

   4. The derivative of sine is cosine:

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

   5. 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:

      $$7 \cos{\left(7 x \right)}$$

   The result is: $10 x^{4} + 7 \cos{\left(7 x \right)} + 4$

The answer is:

$$10 x^{4} + 7 \cos{\left(7 x \right)} + 4$$