Final answer:
The derivative of the function using the definition of derivative. f(x)=1/ˣ²-16f (x) is (1/(x³)) + (16/x²) - (1/(x²)) - 16f(x).
Step-by-step explanation:
To find the derivative of the function using the definition of derivative, we can start by rewriting the given function as f(x) = (1/x²) - 16 f(x).
Next, we need to apply the definition of derivative which states that the derivative of a function f at a point x is equal to the limit as h approaches 0 of (f(x+h) - f(x))/h.
In this case, we need to find the limit of ((1/(x+h)²) - 16f(x+h) - (1/x²) - 16f(x))/h as h approaches 0.
When we simplify the expression, we get (1/(x³)) + (16/x²) - (1/(x²)) - 16f(x).
Taking the limit as h approaches 0, all the terms involving h will cancel out, leaving us with (1/(x³)) + (16/x²) - (1/(x²)) - 16f(x).
This is the derivative of f(x) using the definition of derivative. Therefore the derivate is (1/(x³)) + (16/x²) - (1/(x²)) - 16f(x).