Derivative of 1/(sin(5x))^1/4

1. Apply the quotient rule, which is:

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

   $f{\left(x \right)} = 1$ and $g{\left(x \right)} = \sqrt[4]{\sin{\left(5 x \right)}}$.

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

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

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

   1. Let $u = \sin{\left(5 x \right)}$.

   2. Apply the power rule: $\sqrt[4]{u}$ goes to $\frac{1}{4 u^{\frac{3}{4}}}$

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

      1. Let $u = 5 x$.

      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 $\frac{d}{d x} 5 x$:

         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: $5$

         The result of the chain rule is:

         $$5 \cos{\left(5 x \right)}$$

      The result of the chain rule is:

      $$\frac{5 \cos{\left(5 x \right)}}{4 \sin^{\frac{3}{4}}{\left(5 x \right)}}$$

   Now plug in to the quotient rule:

   $- \frac{5 \cos{\left(5 x \right)}}{4 \sin^{\frac{5}{4}}{\left(5 x \right)}}$

The answer is:

$$- \frac{5 \cos{\left(5 x \right)}}{4 \sin^{\frac{5}{4}}{\left(5 x \right)}}$$