find the derivative
f(x)= (2^x-e^x)/(2^x+e^x)
To find the derivative of the given function f(x) = (2^x - e^x) / (2^x + e^x), we can use the quotient rule.
The quotient rule states that if you have a function in the form f(x) = h(x) / g(x), then the derivative of f(x) with respect to x is given by:
f'(x) = (g(x) h'(x) - h(x) g'(x)) / (g(x))^2
Let's apply the quotient rule to find the derivative of f(x).
First, we need to find h(x) and g(x) based on the given function:
h(x) = 2^x - e^x
g(x) = 2^x + e^x
Next, we calculate the derivatives of h(x) and g(x) separately.
To find h'(x), the derivative of h(x), we use the power rule and chain rule:
h'(x) = (ln(2) * 2^x) - e^x
To find g'(x), the derivative of g(x), we use the power rule and chain rule:
g'(x) = (ln(2) * 2^x) + e^x
Now we can substitute the values into the quotient rule formula:
f'(x) = [(g(x) * h'(x)) - (h(x) * g'(x))] / (g(x))^2
Substituting the values:
f'(x) = [(2^x + e^x) * ((ln(2) * 2^x) - e^x) - ((2^x - e^x) * ((ln(2) * 2^x) + e^x))] / ((2^x + e^x))^2
Simplifying further, we can expand and collect like terms:
f'(x) = [(2^x + e^x) * (ln(2) * 2^x - e^x) - (2^x - e^x) * (ln(2) * 2^x + e^x)] / (2^x + e^x)^2
This is the derivative of f(x) with respect to x.