Derivative of 1/sqrtsin5x

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

2. Apply the power rule: $\frac{1}{u}$ goes to $- \frac{1}{u^{2}}$

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

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

   2. Apply the power rule: $\sqrt{u}$ goes to $\frac{1}{2 \sqrt{u}}$

   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:

      $$\frac{5 \cos{\left(5 x \right)}}{2 \sqrt{\sin{\left(5 x \right)}}}$$

   The result of the chain rule is:

   $$- \frac{5 \cos{\left(5 x \right)}}{2 \sin^{\frac{3}{2}}{\left(5 x \right)}}$$

The answer is: