Mathematic Problems

256. példa

y = \ln \frac{5 x}{4 x - 3}

y = \ln \frac{5 x}{4 x - 3}

g = \frac{5 x}{4 x - 3} \Rightarrow y = \ln g

y' = \frac{1}{g} \cdot g'

g = \frac{5 x}{4 x - 3} = \frac{u}{v}

u = 5 x; u' = 5; v = 4 x - 3; v' = 4

g' = \frac{u' v - u v'}{v^{2}} = \frac{5 \cdot (4 x - 3) - 5 x \cdot 4}{(4 x - 3)^{2}} = \frac{20 x - 15 - 20 x}{{(4 x - 3)}^{2}} = \frac{- 15}{{(4 x - 3)}^{2}}

y' = \frac{1}{g} \cdot g' = \frac{1}{\frac{5 x}{4 x - 3}} \cdot \frac{- 15}{{(4 x - 3)}^{2}}

y' = \frac{4 x - 3}{5 x} \cdot \frac{- 15}{{(4 x - 3)}^{2}}

y' = \frac{- 3}{x \cdot (4 x - 3)}

y' = \frac{- 3}{x \cdot (4 x - 3)}

Matematika - Képlettár