Whats the 2nd derivative of
4xsecx
I got 4secxtanx+8xsecxtan^2x
product rule
4xsec(x)tan(x)+4sec(x)
4(xsec(x)tan(x)+sec(x))
To find the second derivative of 4xsec(x), we need to differentiate it twice.
First, let's find the first derivative using the product rule:
f(x) = 4x * sec(x)
f'(x) = (4 * sec(x)) * x' + (4x) * sec(x)'
Applying the product rule to the first term (4x * sec(x)):
f'(x) = (4 * sec(x)) + (4x) * sec(x) * tan(x)
Now, let's differentiate f'(x) to find the second derivative. We'll apply the product rule again:
f''(x) = ((4 * sec(x)) + (4x) * sec(x) * tan(x))' + (4x * sec(x) * tan(x))'
Differentiating the first term ((4 * sec(x)) + (4x) * sec(x) * tan(x)):
f''(x) = (0 + (4x * sec(x) * tan(x))') + (4x * sec(x) * tan(x))'
The derivative of (4x * sec(x) * tan(x)) is:
[(4x)' * sec(x) * tan(x)] + [4x * (sec(x) * tan(x))']
Since (4x)' = 4 and applying the product rule to the second term (sec(x) * tan(x)):
f''(x) = (4 * sec(x) * tan(x)) + (4x * (sec(x) * tan(x))' + [4x * (sec(x) * tan(x))']'
Simplifying the expression:
f''(x) = 4sec(x)tan(x) + 4x(sec(x)tan'(x) + sec'(x)tan(x)) + 4x(sec(x)tan'(x) + sec'(x)tan(x))'
Now, we can simplify further:
f''(x) = 4sec(x)tan(x) + 4x(sec(x)tan'(x) + sec'(x)tan(x)) + 4x(sec(x)tan'(x) + sec'(x)tan(x))'
Since tan'(x) = sec^2(x), and sec'(x) = sec(x)tan(x),
we can substitute these values into the equation:
f''(x) = 4sec(x)tan(x) + 4x(sec(x)sec^2(x) + sec(x)tan^2(x)) + 4x(sec(x)sec^2(x) + sec(x)tan^2(x))'
Simplifying further:
f''(x) = 4sec(x)tan(x) + 4x(sec^3(x) + sec(x)tan^2(x)) + 4x(sec^3(x) + sec(x)tan^2(x))'
Finally, we have the second derivative of 4xsec(x):
f''(x) = 4sec(x)tan(x) + 4x(sec^3(x) + sec(x)tan^2(x)) + 4x(sec^3(x) + sec(x)tan^2(x))'
To find the second derivative of the function 4xsec(x), we need to use the quotient rule for differentiation.
The quotient rule states that if we have a function in the form of f(x) = g(x) / h(x), then the second derivative can be found using the formula:
f''(x) = [h(x)g''(x) - g(x)h''(x)] / [h(x)]^2
Let's find the first and second derivatives of the numerator and denominator separately:
Numerator (g(x)):
g(x) = 4x
g'(x) = 4 (the derivative of x is 1, and sec(x) doesn't affect the derivative)
g''(x) = 0 (since g(x) = 4x, its second derivative is 0)
Denominator (h(x)):
h(x) = sec(x)
h'(x) = sec(x)tan(x) (using the product rule where the first factor is 1 and the second factor is sec(x))
h''(x) = sec(x)tan^2(x) + sec^3(x) (using the product rule once again)
Now we can substitute these values into the formula for the second derivative:
f''(x) = [h(x)g''(x) - g(x)h''(x)] / [h(x)]^2
= [sec(x)(0) - (4x)(sec(x)tan^2(x) + sec^3(x))] / [sec(x)]^2
= [-4xsec(x)tan^2(x) - 4sec^3(x)] / sec^2(x)
Now, let's simplify this expression further:
f''(x) = -4xsec(x)tan^2(x) / sec^2(x) - 4sec^3(x) / sec^2(x)
= -4xtan^2(x) - 4sec(x)
Therefore, the second derivative of 4xsec(x) is -4xtan^2(x) - 4sec(x).