Final answer:
The derivative of f(t) = sec(t)(sec(t) tan(t)) is found using the product rule and yields f'(t) = sec^2(t)tan(t) + sec(t)(sec^3(t) + sec(t)tan^2(t)). Further simplification would involve trigonometric identities.
Step-by-step explanation:
To find the derivative of f(t) = sec(t)(sec(t) tan(t)), we apply the product rule since this is a product of two functions of t. The product rule states that the derivative of two functions v(t) and u(t) is given by v'(t)u(t) + v(t)u'(t). In this case, we treat sec(t) as v(t) and sec(t)tan(t) as u(t).
First, find the derivatives separately:
- The derivative of sec(t) is sec(t)tan(t).
- The derivative of sec(t)tan(t) can be found by applying the product rule again, yielding sec(t)sec^2(t) + tan^2(t)sec(t).
Now, applying the product rule to the original function:
f'(t) = sec(t)tan(t)sec(t) + sec(t)(sec(t)sec^2(t) + tan^2(t)sec(t))
We can simplify this expression by factoring out common terms and using trigonometric identities.