Derivative of (5*x-4)*(2*x^4-7*x+1)

1. Apply the product rule:

   $$\frac{d}{d x} f{\left(x \right)} g{\left(x \right)} = f{\left(x \right)} \frac{d}{d x} g{\left(x \right)} + g{\left(x \right)} \frac{d}{d x} f{\left(x \right)}$$

   $f{\left(x \right)} = 5 x - 4$; to find $\frac{d}{d x} f{\left(x \right)}$:

   1. Differentiate $5 x - 4$ 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: $x$ goes to $1$

         So, the result is: $5$

      2. The derivative of the constant $-4$ is zero.

      The result is: $5$

   $g{\left(x \right)} = \left(2 x^{4} - 7 x\right) + 1$; to find $\frac{d}{d x} g{\left(x \right)}$:

   1. Differentiate $\left(2 x^{4} - 7 x\right) + 1$ term by term:

      1. Differentiate $2 x^{4} - 7 x$ 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: $x^{4}$ goes to $4 x^{3}$

            So, the result is: $8 x^{3}$

         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: $8 x^{3} - 7$

      2. The derivative of the constant $1$ is zero.

      The result is: $8 x^{3} - 7$

   The result is: $\left(5 x - 4\right) \left(8 x^{3} - 7\right) + 5 \left(2 x^{4} - 7 x\right) + 5$

2. Now simplify:

   $$50 x^{4} - 32 x^{3} - 70 x + 33$$

The answer is:

$$50 x^{4} - 32 x^{3} - 70 x + 33$$