By direct use of the definition of a derivative, the derivative of
f(x) = 1/x
?
1/(x+h) - 1/x
(x - x - h)/(x^2+xh)
-h/(x^2+xh)
divide by h
-1/(x^2+xh)
let h--->0
-1/x^2 sure enough
To find the derivative of the function f(x) = 1/x using the definition of a derivative, we need to apply the limit definition of the derivative.
The derivative of a function f(x) at a point x is defined as:
f'(x) = lim(h -> 0) [f(x + h) - f(x)] / h
In this case, we have:
f(x) = 1/x
To find the derivative, we substitute f(x + h) and f(x) into the definition, and simplify the expression:
f'(x) = lim(h -> 0) [1/(x + h) - 1/x] / h
Next, we need to simplify the expression by finding a common denominator:
f'(x) = lim(h -> 0) [(x - (x + h))/(x(x + h))] / h
Simplifying further:
f'(x) = lim(h -> 0) [-h / (x(x + h))] / h
The h terms on the numerator and denominator cancel out:
f'(x) = lim(h -> 0) -1 / (x(x + h))
Now, we can take the limit as h approaches 0:
f'(x) = -1 / (x^2)
Therefore, the derivative of f(x) = 1/x by direct use of the definition of a derivative is f'(x) = -1 / (x^2).