Derivative of z=y4sin5x+2x−2

1. Differentiate $\left(2 x + y_{4} \sin{\left(5 x \right)}\right) - 2$ term by term:

   1. Differentiate $2 x + y_{4} \sin{\left(5 x \right)}$ term by term:

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

         1. Let $u = 5 x$.

         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 $\frac{d}{d x} 5 x$:

            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:

            $$5 \cos{\left(5 x \right)}$$

         So, the result is: $5 y_{4} \cos{\left(5 x \right)}$

      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: $2$

      The result is: $5 y_{4} \cos{\left(5 x \right)} + 2$

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

   The result is: $5 y_{4} \cos{\left(5 x \right)} + 2$

The answer is:

$$5 y_{4} \cos{\left(5 x \right)} + 2$$