Find the second derivative of the function.
g(x)=x^6secx
g ' (x) = x^6 (secxtanx) + 6x^5 secx
g '' (x) = x^6[secx secx tanx + tanx sec^2 x)
+ secx [ x^6 sec^2 x + 6x^5 tanx] + tanx [ x^6 secxtanx + 6x^5 secx] + 6x^5 secxtanx + 30x^4 secx
Thank You so much for all the answers(:
To find the second derivative of the function g(x) = x^6sec(x), follow these steps:
Step 1: Find the first derivative of g(x) with respect to x.
The first derivative of g(x) can be found using the product rule. Let's call the first factor f(x) = x^6 and the second factor h(x) = sec(x).
Using the product rule: (f(x)h(x))' = f'(x)h(x) + f(x)h'(x)
f'(x) = 6x^5 (the derivative of x^6 with respect to x)
h'(x) = sec(x)tan(x) (the derivative of sec(x) with respect to x)
Applying the product rule: g'(x) = f'(x)h(x) + f(x)h'(x)
g'(x) = (6x^5)sec(x) + (x^6)(sec(x)tan(x))
g'(x) = 6x^5sec(x) + x^6sec(x)tan(x)
Step 2: Find the second derivative of g(x) with respect to x.
To find the second derivative, we need to differentiate g'(x) with respect to x.
Using the product rule again:
g''(x) = (6x^5sec(x) + x^6sec(x)tan(x))'
= (6x^5sec(x))' + (x^6sec(x)tan(x))'
Differentiating each term separately:
(6x^5sec(x))' = 30x^4sec(x) + 6x^5sec(x)tan(x)
(x^6sec(x)tan(x))' = 6x^5sec(x)tan(x) + x^6sec(x)tan^2(x)
Combining these results, we have:
g''(x) = 30x^4sec(x) + 6x^5sec(x)tan(x) + 6x^5sec(x)tan(x) + x^6sec(x)tan^2(x)
= 30x^4sec(x) + 12x^5sec(x)tan(x) + x^6sec(x)tan^2(x)
Therefore, the second derivative of g(x) = x^6sec(x) is g''(x) = 30x^4sec(x) + 12x^5sec(x)tan(x) + x^6sec(x)tan^2(x).