Derivative of 5sin7x-7x²+7

1. Differentiate $- 7 x^{2} + 5 \sin{\left(7 x \right)} + 7$ 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 sine is cosine:

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

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

         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:

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

      So, the result is: $35 \cos{\left(7 x \right)}$

   2. 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: $x^{2}$ goes to $2 x$

         So, the result is: $14 x$

      So, the result is: $- 14 x$

   3. The derivative of the constant $7$ is zero.

   The result is: $- 14 x + 35 \cos{\left(7 x \right)}$

The answer is:

$$- 14 x + 35 \cos{\left(7 x \right)}$$