Derivative of cos(5x/2)-7x+3

1. Differentiate $\left(- 7 x + \cos{\left(\frac{5 x}{2} \right)}\right) + 3$ term by term:

   1. Differentiate $- 7 x + \cos{\left(\frac{5 x}{2} \right)}$ term by term:

      1. Let $u = \frac{5 x}{2}$.

      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 $\frac{d}{d x} \frac{5 x}{2}$:

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

            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: $5$

            So, the result is: $\frac{5}{2}$

         The result of the chain rule is:

         $$- \frac{5 \sin{\left(\frac{5 x}{2} \right)}}{2}$$

      4. 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 is: $- \frac{5 \sin{\left(\frac{5 x}{2} \right)}}{2} - 7$

   2. The derivative of the constant $3$ is zero.

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

2. Now simplify:

   $$- \frac{5 \sin{\left(\frac{5 x}{2} \right)}}{2} - 7$$

The answer is:

$$- \frac{5 \sin{\left(\frac{5 x}{2} \right)}}{2} - 7$$