Final answer:
To write derivatives in Taylor form, calculate the derivatives at a point 'a' and use them in the Taylor series expansion of the function around that point. Substitute the computed derivatives into the series to obtain the Taylor form.
Step-by-step explanation:
To write derivatives in Taylor form, you use the Taylor series expansion. The Taylor series of a function f(x) about a point a is given by:
- f(a)
- + f'(a)(x - a)
- + \(rac{f''(a)}{2!}\)(x - a)2
- + \(rac{f'''(a)}{3!}\)(x - a)3
- + ... +
- + \(rac{f(n)(a)}{n!}\)(x - a)n + ...
where f'(a), f''(a), ..., f(n)(a) are the first, second, ..., nth derivatives of f at the point a, respectively, and n! denotes the factorial of n.
To find these derivatives, you apply the rules of differentiation to the function. For example, if the function is f(x) = x2, the first derivative is 2x, and the second derivative is 2. You substitute these derivatives into the Taylor series expansion at the point a to get the Taylor form of the derivative.