Final answer:
To find the derivative of ln(tanx+secx), we apply the chain rule. The derivative is (1/(tanx+secx))*(sec^2(x)+tan(x)*sec(x)).
Step-by-step explanation:
To find the derivative of ln(tanx+secx), we can use the chain rule. Let's denote the function as f(x) = ln(tanx+secx). The chain rule states that if we have a composite function f(g(x)), the derivative is given by f'(g(x)) * g'(x).
Applying the chain rule, we have:
f'(x) = (1/(tanx+secx))*(sec^2(x)+tan(x)*sec(x))
Apply the chain rule: [ \frac{d}{dx}[\ln(\tan x + \sec x)] = \frac{1}{\tan x + \sec x} \cdot (\sec^2 x + \sec x \tan x) ].
Therefore, the derivative of ln(tanx+secx) is (1/(tanx+secx))*(sec^2(x)+tan(x)*sec(x)).
So, the derivative is: [ \frac{\sec^2 x + \sec x \tan x}{\tan x + \sec x} ].