if f(x) = √(1+x^3) find the 2nd derivative
To find the second derivative of f(x), we need to first find the first derivative f'(x) and then take the derivative of f'(x). Let's start with the first derivative:
f'(x) = d/dx [√(1+x^3)]
= d/dx [(1+x^3)^(1/2)]
Using the chain rule, we have:
f'(x) = (1/2)(1+x^3)^(-1/2) * (3x^2)
= (3x^2) / (2 * √(1+x^3))
Now, let's find the second derivative f''(x):
f''(x) = d/dx [(3x^2) / (2 * √(1+x^3))]
We can rewrite the expression as:
f''(x) = d/dx [(3x^2)(1+x^3)^(-1/2) / 2]
Applying the product rule:
f''(x) =[ (d/dx(3x^2)) * (1+x^3)^(-1/2)/2] + [(3x^2) * d/dx[(1+x^3)^(-1/2)/2]]
Taking the derivatives:
f''(x) = [6x * (1+x^3)^(-1/2)/2] + [(3x^2) * (-1/2) * (1+x^3)^(-3/2) * (3x^2) * 1/2]
Now simplifying:
f''(x) = (3x(1+x^3)^(-1/2)) - (9x^4(1+x^3)^(-3/2))
To make it neater, we can rewrite it as:
f''(x) = (3x/(1+x^3)^(1/2)) - (9x^4/(1+x^3)^(3/2))
Finally, we can rewrite the expression using a common denominator:
f''(x) = [3x(1+x^3) - 9x^4] / (1+x^3)^(3/2).