Question

Find the third derivative of f(x)=∛6x+1
f^(4)(x)=

Answer 1

`f(x) = \sqrt[3]{6x+1} = ( 6x +1 )^(\tfrac 13)\\\\\\f'(x) = \frac 13 (6x +1)^(\tfrac 13 - 1) \cdot (6+0) = 2 (6x+1)^{-\tfrac{2}3}\\\\\\f''(x) = 2 \left( -\frac 23 \right) (6x+1)^(-\tfrac 23 - 1) \cdot(6+0)= -8(6x+1)^(-\tfrac 53)\\\\\\f'''(x) = -8 \left( - \frac 53\right) (6x+1)^(-\tfrac 53 - 1) \cdot (6+0) = 80(6x+1)^(-\tfrac 83)\\\\\text{Hence the third derivative of f(x) is}~ 80(6x+1)^(-\tfrac 83)\\`