How do you find the derivative of $5 ln (7 x + 6 ln (x))$ $5ln(7x+6ln(x))$ ?

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How do you find the derivative of $5 ln (7 x + 6 ln (x))$ $5ln(7x+6ln(x))$ ?

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Answer 1 · 2015-05-25T19:17:15

SImple: you use the chain rule!

The chain rule states that

$\frac{d y}{d x} = \frac{d y}{d u} \frac{d u}{d x}$ $(dy)/(dx)=(dy)/(du)(du)/(dx)$

Thus, we just need to rename $u = 7 x + 6 ln (x)$ $u=7x+6ln(x)$ (consequently our original function becomes $5 ln (u)$ $5ln(u)$ ), and now derivate it all part by part:

$\frac{d y}{d u} = 5 \cdot \frac{1}{u}$ $(dy)/(du)=5*1/u$

$\frac{d u}{d x} = 7 + 6 \cdot \frac{1}{x}$ $(du)/(dx)=7+6*1/x$

Aggregating them:

$\frac{d y}{d x} = \frac{5}{u} (7 + \frac{6}{x}) = \frac{5}{7 x + 6 ln (x)} (7 + \frac{6}{x}) = \frac{35 + \frac{30}{x}}{7 x + 6 ln (x)}$ $(dy)/(dx)=5/u(7+6/x)=5/(7x+6ln(x))(7+6/x)=color(green)((35+30/x)/(7x+6ln(x)))$

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How do you find the derivative of $5 ln (7 x + 6 ln (x))$ $5ln(7x+6ln(x))$ ?

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How do you find the derivative of $5 ln (7 x + 6 ln (x))$ $5ln(7x+6ln(x))$ ?