Power Rule in Differentiation

Power Rule

The power rule is a fundamental rule in differentiation that allows us to differentiate functions of the form:

Formula:

If f(x) = x^n, where n is a real number, then f'(x) = nx^(n-1)

Key Points:

• The power rule can be applied to any function that can be written in the form x^n, where n is a constant.
• The derivative of x^n is n times x to the power of n-1.
• The power rule can be applied recursively to find higher-order derivatives.

Examples:

• If f(x) = x^2, then f'(x) = 2x^(2-1) = 2x
• If f(x) = x^3, then f'(x) = 3x^(3-1) = 3x^2
• If f(x) = x^(-2), then f'(x) = (-2)x^(-2-1) = -2x^(-3)

Special Cases:

• If f(x) = x, then f'(x) = 1x^(1-1) = 1
• If f(x) = x^0, then f'(x) = 0x^(0-1) = 0 (since any number to the power of 0 is 1, and the derivative of 1 is 0)

Applications:

• The power rule is used to find the derivative of polynomial functions, which are functions of the form f(x) = a_nx^n + a_(n-1)x^(n-1) + ... + a_1x + a_0.
• The power rule is also used to find the derivative of functions that involve rational powers, such as f(x) = x^(1/2) or f(x) = x^(-3/2).

Power Rule

Formula

• If f(x) = x^n, where n is a real number, then f'(x) = nx^(n-1).

Key Points

• The power rule can be applied to any function that can be written in the form x^n, where n is a constant.
• The derivative of x^n is n times x to the power of n-1.
• The power rule can be applied recursively to find higher-order derivatives.

Examples

• If f(x) = x^2, then f'(x) = 2x^(2-1) = 2x.
• If f(x) = x^3, then f'(x) = 3x^(3-1) = 3x^2.
• If f(x) = x^(-2), then f'(x) = (-2)x^(-2-1) = -2x^(-3).

Special Cases

• If f(x) = x, then f'(x) = 1x^(1-1) = 1.
• If f(x) = x^0, then f'(x) = 0x^(0-1) = 0.

Applications

• The power rule is used to find the derivative of polynomial functions, which are functions of the form f(x) = a_nx^n + a_(n-1)x^(n-1) +...+ a_1x + a_0.
• The power rule is also used to find the derivative of functions that involve rational powers, such as f(x) = x^(1/2) or f(x) = x^(-3/2).