Question

Differentiating a Logarithmic Function in Exercise, find the derivative of the function. See Examples 1, 2, 3, and 4.

g(x) = ln {e^x + e^-x/2}

Answer 1

`g'(x)=\frac{2e^x-e^{-(x)/(2)}}{2(e^x+e^{-(x)/(2)})}`

Explanation:

We are given that a function

`g(x)=ln(e^x+e^{-(x)/(2)})`

We have to find the derivative of function

Differentiate w.r.t x

`g'(x)=\frac{1}{e^x+e^{-(x)/(2)}}* (e^x+e^{-(x)/(2)}* (-(1)/(2)))`

By using formula

`(d(lnx))/(dx)=(1)/(x)`

`(d e^x)/(dx)=e^x`

`g'(x)=\frac{e^x-\frac{e^{-(x)/(2)}}{2}}{e^x+e^{-(x)/(2)}}`

Hence, the derivative function

`g'(x)=\frac{2e^x-e^{-(x)/(2)}}{2(e^x+e^{-(x)/(2)})}`