If y = log left[e^{5x} left(frac{3x-4}{x+5}ri | vedclass

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(A) Given $y = \log \left[e^{5x} \left(\frac{3x-4}{x+5}\right)^{\frac{4}{3}}\right]$.Using the properties of logarithms,$\log(ab) = \log a + \log b$ a

Vedclass · Accepted Answer

$5 + \frac{4}{3x-4} - \frac{4}{3(x+5)}$

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(A) Given $y = \log \left[e^{5x} \left(\frac{3x-4}{x+5}\right)^{\frac{4}{3}}\right]$.Using the properties of logarithms,$\log(ab) = \log a + \log b$ and $\log(a^n) = n \log a$:$y = \log(e^{5x}) + \log\left(\left(\frac{3x-4}{x+5}\right)^{\frac{4}{3}}\right)$$y = 5x \log e + \frac{4}{3} \log\left(\frac{3x-4}{x+5}\right)$$y = 5x + \frac{4}{3} [\log(3x-4) - \log(x+5)]$Now,differentiate with respect to $x$:$\frac{dy}{dx} = \frac{d}{dx}(5x) + \frac{4}{3} \left[ \frac{d}{dx}(\log(3x-4)) - \frac{d}{dx}(\log(x+5)) \right]$$\frac{dy}{dx} = 5 + \frac{4}{3} \left[ \frac{1}{3x-4} \cdot 3 - \frac{1}{x+5} \cdot 1 \right]$$\frac{dy}{dx} = 5 + \frac{4}{3x-4} - \frac{4}{3(x+5)}$.

If $y = \log \left[e^{5x} \left(\frac{3x-4}{x+5}\right)^{\frac{4}{3}}\right]$,then $\frac{dy}{dx}$ is equal to

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