Question

if f(x) = √(1+x^3) find the 2nd derivative

Answer 1

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).