Derivative of y=ln(4x+5x2)1/x

1. Apply the quotient rule, which is:

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

   $f{\left(x \right)} = \log{\left(4 x + 5 x_{2} \right)}$ and $g{\left(x \right)} = x$.

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

   1. Let $u = 4 x + 5 x_{2}$.

   2. The derivative of $\log{\left(u \right)}$ is $\frac{1}{u}$.

   3. Then, apply the chain rule. Multiply by $\frac{\partial}{\partial x} \left(4 x + 5 x_{2}\right)$:

      1. Differentiate $4 x + 5 x_{2}$ 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: $4$

         2. The derivative of the constant $5 x_{2}$ is zero.

         The result is: $4$

      The result of the chain rule is:

      $$\frac{4}{4 x + 5 x_{2}}$$

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

   1. Apply the power rule: $x$ goes to $1$

   Now plug in to the quotient rule:

   $\frac{\frac{4 x}{4 x + 5 x_{2}} - \log{\left(4 x + 5 x_{2} \right)}}{x^{2}}$

2. Now simplify:

   $$\frac{4 x - \left(4 x + 5 x_{2}\right) \log{\left(4 x + 5 x_{2} \right)}}{x^{2} \cdot \left(4 x + 5 x_{2}\right)}$$

The answer is:

$$\frac{4 x - \left(4 x + 5 x_{2}\right) \log{\left(4 x + 5 x_{2} \right)}}{x^{2} \cdot \left(4 x + 5 x_{2}\right)}$$