Find the second derivate d^2y/dx^2 when y = x^6 + sqrt(x).
Initially we find the first derivative of the function y = x6 + sqrt(x). We achieve this by multiplying each x term by the power it is raised to, then reducing the power by 1. Solution:
1) It helps to initially simplify the sqrt(x) term to x1/2 to give: y = x6 + x1/2
2) We can then determine the first derivative: dy/dx = 6x5+ 1/2x-1/2
To determine the second derivative we then take the first derivative and differentiate that function, repeating the prior steps:
3) d2y/dx2 = 30x4 + (-1/4)x-3/2
We can simplify the answer to give:
4) d2y/dx2 = 30x4 -1/4x-3/2
Simplifying fully gives:
5) d2y/dx2 = 30x4 - (1/(4x3/2))
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