Differential Calculus I: Definition: A differentiation rule is a formula that can be used to find the derivative of a function. A derivative measures ho...
Differential Calculus I:
Definition: A differentiation rule is a formula that can be used to find the derivative of a function. A derivative measures how quickly a function changes with respect to its input.
The basic differentiation rules are:
Power rule: If f(x) = x^n, then f'(x) = nx^(n-1).
Constant rule: If f(x) = a, then f'(x) = 0.
Sum rule: If f(x) = g(x) + h(x), then f'(x) = g'(x) + h'(x).
Difference rule: If f(x) = g(x) - h(x), then f'(x) = g'(x) - h'(x).
Examples:
Power rule: If f(x) = x^2, then f'(x) = 2x.
Constant rule: If f(x) = 5, then f'(x) = 0.
Sum rule: If f(x) = x^3 + 2x, then f'(x) = 3x^2 + 2.
Difference rule: If f(x) = x^2 - 3x + 1, then f'(x) = 2x - 3.
Importance: Differentiation rules help us to find the rate of change of a function, which is useful in various applications, such as physics, economics, and finance. They also help us to differentiate composite functions, which are functions of other functions