Derivative of (5x-7)^8+tg7x

1. Differentiate $\left(5 x - 7\right)^{8} + \tan{\left(7 x \right)}$ term by term:

   1. Let $u = 5 x - 7$.

   2. Apply the power rule: $u^{8}$ goes to $8 u^{7}$

   3. Then, apply the chain rule. Multiply by $\frac{d}{d x} \left(5 x - 7\right)$:

      1. Differentiate $5 x - 7$ 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 $-7$ is zero.

         The result is: $5$

      The result of the chain rule is:

      $$40 \left(5 x - 7\right)^{7}$$

   4. Rewrite the function to be differentiated:

      $$\tan{\left(7 x \right)} = \frac{\sin{\left(7 x \right)}}{\cos{\left(7 x \right)}}$$

   5. Apply the quotient rule, which is:

      $$\frac{d}{d x} \frac{f{\left(x \right)}}{g{\left(x \right)}} = \frac{- f{\left(x \right)} \frac{d}{d x} g{\left(x \right)} + g{\left(x \right)} \frac{d}{d x} f{\left(x \right)}}{g^{2}{\left(x \right)}}$$

      $f{\left(x \right)} = \sin{\left(7 x \right)}$ and $g{\left(x \right)} = \cos{\left(7 x \right)}$.

      To find $\frac{d}{d x} f{\left(x \right)}$:

      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:

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

      To find $\frac{d}{d x} g{\left(x \right)}$:

      1. Let $u = 7 x$.

      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 $\frac{d}{d x} 7 x$:

         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:

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

      Now plug in to the quotient rule:

      $\frac{7 \sin^{2}{\left(7 x \right)} + 7 \cos^{2}{\left(7 x \right)}}{\cos^{2}{\left(7 x \right)}}$

   The result is: $40 \left(5 x - 7\right)^{7} + \frac{7 \sin^{2}{\left(7 x \right)} + 7 \cos^{2}{\left(7 x \right)}}{\cos^{2}{\left(7 x \right)}}$

2. Now simplify:

   $$3125000 x^{7} - 30625000 x^{6} + 128625000 x^{5} - 300125000 x^{4} + 420175000 x^{3} - 352947000 x^{2} + 164708600 x + 7 \tan^{2}{\left(7 x \right)} - 32941713$$

The answer is:

$$3125000 x^{7} - 30625000 x^{6} + 128625000 x^{5} - 300125000 x^{4} + 420175000 x^{3} - 352947000 x^{2} + 164708600 x + 7 \tan^{2}{\left(7 x \right)} - 32941713$$