Derivative of f(x)=(6x²+7x)⁴

1. Let $u = 6 x^{2} + 7 x$.

2. Apply the power rule: $u^{4}$ goes to $4 u^{3}$

3. Then, apply the chain rule. Multiply by $\frac{d}{d x} \left(6 x^{2} + 7 x\right)$:

   1. Differentiate $6 x^{2} + 7 x$ 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: $x^{2}$ goes to $2 x$

         So, the result is: $12 x$

      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: $7$

      The result is: $12 x + 7$

   The result of the chain rule is:

   $$4 \cdot \left(12 x + 7\right) \left(6 x^{2} + 7 x\right)^{3}$$

4. Now simplify:

   $$x^{3} \left(6 x + 7\right)^{3} \cdot \left(48 x + 28\right)$$

The answer is:

$$x^{3} \left(6 x + 7\right)^{3} \cdot \left(48 x + 28\right)$$