Gradient descent maximizes a function using knowledge of its derivative. Newton's method, a root finding algorithm, maximizes a function using knowledge of its second derivative. That can be faster when the second derivative is known and easy to compute (the Newton-Raphson algorithm is used in logistic regression). However, the analytic expression for the second derivative is often complicated or intractable, requiring a lot of computation. Numerical methods for computing the second derivative also require a lot of computation -- if $N$ values are required to compute the first derivative  ,$N^2$ are required for the second derivative.

Gradient descent maximizes a function using knowledge of its derivative. Newton's method, a root finding algorithm, maximizes a function using knowledge of its second derivative. That can be faster when the second derivative is known and easy to compute (the Newton-Raphson algorithm is used in logistic regression). However, the analytic expression for the second derivative is often complicated or intractable, requiring a lot of computation. Numerical methods for computing the second derivative also require a lot of computation -- if $N$ values are required to compute the first derivative,  $N^2$ are required for the second derivative.