Final answer:
The derivative of f(x) = sec(x)/x is found using the quotient rule resulting in (x sec(x) tan(x) - sec(x)) / x^2. The second derivative of f(x) = 4x^(3/2) uses the power rule, leading to a final answer of 6x^(-1/2).
Step-by-step explanation:
The derivative of the trigonometric function f(x) = sec(x)/x requires the application of the quotient rule, which is used when differentiating ratios of functions. The derivative of sec(x) is sec(x)tan(x), and the derivative of x is simply 1. Applying the quotient rule, which is (v(u') - u(v')) / v^2 for a function u/v, we find:
f'(x) = (x(sec(x)tan(x)) - sec(x)(1))/(x^2) = (x sec(x) tan(x) - sec(x)) / x^2
For the second derivative of f(x) = 4x^(3/2), we will use the power rule for derivatives, which states that the derivative of x^n is n*x^(n-1). For this function:
f'(x) = rac{d}{dx}(4x^(3/2)) = 4*rac{3}{2}*x^(1/2)
The second derivative is then:
f''(x) = rac{d}{dx}(4*rac{3}{2}*x^(1/2)) = 4*rac{3}{2}*rac{1}{2}*x^(-1/2) = 6x^(-1/2)