Derivative of (3*tan(x)+5)*x^7

1. Apply the product rule:

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

   $f{\left(x \right)} = 3 \tan{\left(x \right)} + 5$; to find $\frac{d}{d x} f{\left(x \right)}$:

   1. Differentiate $3 \tan{\left(x \right)} + 5$ term by term:

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

         1. Rewrite the function to be differentiated:

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

         2. Apply the quotient rule, which is:

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

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

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

            1. The derivative of sine is cosine:

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

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

            1. The derivative of cosine is negative sine:

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

            Now plug in to the quotient rule:

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

         So, the result is: $\frac{3 \left(\sin^{2}{\left(x \right)} + \cos^{2}{\left(x \right)}\right)}{\cos^{2}{\left(x \right)}}$

      2. The derivative of the constant $5$ is zero.

      The result is: $\frac{3 \left(\sin^{2}{\left(x \right)} + \cos^{2}{\left(x \right)}\right)}{\cos^{2}{\left(x \right)}}$

   $g{\left(x \right)} = x^{7}$; to find $\frac{d}{d x} g{\left(x \right)}$:

   1. Apply the power rule: $x^{7}$ goes to $7 x^{6}$

   The result is: $\frac{3 x^{7} \left(\sin^{2}{\left(x \right)} + \cos^{2}{\left(x \right)}\right)}{\cos^{2}{\left(x \right)}} + 7 x^{6} \left(3 \tan{\left(x \right)} + 5\right)$

2. Now simplify:

   $$x^{6} \left(\frac{3 x}{\cos^{2}{\left(x \right)}} + 21 \tan{\left(x \right)} + 35\right)$$

The answer is:

$$x^{6} \left(\frac{3 x}{\cos^{2}{\left(x \right)}} + 21 \tan{\left(x \right)} + 35\right)$$