The formula for derivatives
The formula for the derivative of a function is given by the limit of the difference quotient:
f'(x) = lim h -> 0 (f(x + h) – f(x)) / h
- Where f'(x) is the derivative of the function f at the point x, and
- h is a small change in the input x.
The limit represents the rate of change of the function as h approaches zero.
There are many rules and techniques for finding the derivatives of various functions, including algebraic, exponential, logarithmic, and trigonometric functions.
Some common rules for finding derivatives include
- the power rule,
- the product rule,
- the quotient rule,
- the chain rule, and
- the implicit differentiation rule.
For example,
using the power rule, the derivative of a function f(x) = x^n (where n is a constant) is given by:
f'(x) = nx^(n-1)
Using the product rule, the derivative of the product of two functions, f(x) and g(x), is given by:
(f(x)g(x))’ = f'(x)g(x) + f(x)g'(x)
These are just a few examples of the formulas for derivatives.
The formula for the derivative of a function depends on the specific function being differentiated.