Question #3003

what is the derivaties of the following functions :
1- y = 4 / x-5 Inx
2- y = e^-x + e^x / e^x +1
3- y = e^ (1 - In x)
4- y = 6 In ( x^ 1/2 - 3 x)

Expert's answer

<img src="/cgi-bin/mimetex.cgi?1.%20y%20=%20%5Cfrac%7B4%7D%7Bx-5%7D%20%5Cln%7Bx%7D%20%5Chspace%7B3mm%7D%20y%27%20=%20-%5Cfrac%7B4%7D%7B%28x-5%29%5E2%7D%20%5Cln%7Bx%7D%20+%20%5Cfrac%7B4%7D%7Bx%28x-5%29%7D%20%5C%5C%202.%20y%20=%20e%5E%7B-x%7D%20+%20%5Cfrac%7Be%5Ex%7D%7Be%5Ex%20+%201%7D%20=%20e%5E%7B-x%7D%20+%20%5Cfrac%7Be%5Ex%20+%201%20-1%7D%7Be%5Ex%20+1%7D%20=%20e%5E%7B-x%7D%20+%201%20-%20%5Cfrac%7B1%7D%7Be%5Ex%20+%201%7D%20%5C%5C%20%5Chspace%7B6mm%7D%20y%27%20=%20-%20e%5E%7B-x%7D%20+%20%5Cfrac%7Be%5Ex%7D%7B%28e%5Ex+1%29%5E2%7D%20%5C%5C%203.%20y%20=%20e%5E%7B1-%20%5Cln%7Bx%7D%7D%20=%20e/e%5E%7B%5Cln%20x%7D%20=%20e/x%20%5Chspace%7B6mm%7D%20y%27%20=%20-%20e/x%5E2%5C%5C%204.%20y%20=%206%20%5Cln%20%28x%5E%7B1/2%7D%20-%203x%29%20%5Chspace%7B6mm%7D%20y%27%20=%206%20%5Cfrac%7B1/2%20x%5E%7B-1/2%7D%20-3%7D%7Bx%5E%7B1/2%7D%20-%203x%7D" title="1. y = \frac{4}{x-5} \ln{x} \hspace{3mm} y' = -\frac{4}{(x-5)^2} \ln{x} + \frac{4}{x(x-5)} \\ 2. y = e^{-x} + \frac{e^x}{e^x + 1} = e^{-x} + \frac{e^x + 1 -1}{e^x +1} = e^{-x} + 1 - \frac{1}{e^x + 1} \\ \hspace{6mm} y' = - e^{-x} + \frac{e^x}{(e^x+1)^2} \\ 3. y = e^{1- \ln{x}} = e/e^{\ln x} = e/x \hspace{6mm} y' = - e/x^2\\ 4. y = 6 \ln (x^{1/2} - 3x) \hspace{6mm} y' = 6 \frac{1/2 x^{-1/2} -3}{x^{1/2} - 3x}">

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