Whats the 2nd derivative of

4xsecx

I got 4secxtanx+8xsecxtan^2x

To find the second derivative of the function 4xsec(x), you will need to use the quotient rule and the chain rule for differentiation. Let's break it down step by step:

Step 1: Start with the given function
f(x) = 4xsec(x)

Step 2: Rewrite sec(x) as 1/cos(x)
f(x) = 4x * (1/cos(x))

Step 3: Apply the product rule
f'(x) = (4 * d(x)/dx) * (1/cos(x)) + (4x) * (d(1/cos(x))/dx)

Step 4: Simplify and differentiate each term
f'(x) = 4 * (1/cos(x)) + 4x * (-1/cos(x)^2) * (-sin(x))

Step 5: Combine like terms
f'(x) = 4/cos(x) - 4xsin(x)/cos^2(x)
= 4sec(x) - 4xsin(x)/cos^2(x)

Step 6: Now you have the first derivative, f'(x). To find the second derivative, differentiate f'(x) using the product rule and chain rule if necessary. Let's differentiate the first and second terms separately.

Differentiating the first term:
d(4sec(x))/dx = 4(d(sec(x))/dx)
= 4(sec(x)tan(x))

Differentiating the second term:
d(-4xsin(x)/cos^2(x))/dx = -4(sind(x))/cos^2(x) - 4x(d(sin(x))/dx(1/cos^2(x)))
= -4(sind(x))/cos^2(x) - 4x(-cos(x)/cos^2(x))
= -4(sind(x))/cos^2(x) + 4xcos(x)/cos^3(x)

Step 7: Combine the results from step 6
The second derivative, f"(x), is given by adding the differentiated terms:

f"(x) = 4(sec(x)tan(x)) - 4(sind(x))/cos^2(x) + 4xcos(x)/cos^3(x)

Simplifying further is possible, but it depends on whether you need/want specific simplifications.