find the derivative of y with respect to x of ( y = 2 ln left( \frac { 5 } { x } \right) )\nif ( y = c ln u…

find the derivative of y with respect to x of ( y = 2 ln left( \frac { 5 } { x } \right) )\nif ( y = c ln u ), identify u from the given function. then use it to find ( \frac { d u } { d x } )\n( u = \frac { 5 } { x } )\n( \frac { d u } { d x } = - \frac { 5 } { x ^ { 2 } } )\nthe derivative of y with respect to x of ( y = 2 ln left( \frac { 5 } { x } \right) ) is

find the derivative of y with respect to x of ( y = 2 ln left( \frac { 5 } { x } \right) )\nif ( y = c ln u ), identify u from the given function. then use it to find ( \frac { d u } { d x } )\n( u = \frac { 5 } { x } )\n( \frac { d u } { d x } = - \frac { 5 } { x ^ { 2 } } )\nthe derivative of y with respect to x of ( y = 2 ln left( \frac { 5 } { x } \right) ) is

Answer

Explanation:

Step1: Apply the chain rule

The chain rule states that if (y = c\ln u), then (\frac{dy}{dx}=\frac{c}{u}\cdot\frac{du}{dx}). Here (c = 2), (u=\frac{5}{x}), and (\frac{du}{dx}=-\frac{5}{x^{2}}).

Step2: Substitute (u) and (\frac{du}{dx}) into the chain - rule formula

Substitute (u=\frac{5}{x}) and (\frac{du}{dx}=-\frac{5}{x^{2}}) into (\frac{dy}{dx}=\frac{c}{u}\cdot\frac{du}{dx}). We get (\frac{dy}{dx}=\frac{2}{\frac{5}{x}}\cdot\left(-\frac{5}{x^{2}}\right)).

Step3: Simplify the expression

First, (\frac{2}{\frac{5}{x}}=\frac{2x}{5}). Then (\frac{2x}{5}\cdot\left(-\frac{5}{x^{2}}\right)=-\frac{2}{x}).

Answer:

(-\frac{2}{x})