Calculate the derivative of a power function f(x) = axⁿ using the Power Rule.
Function Definition
Input Function
f(x) = 1x2
Derivative Result
f'(x)2x
Calculation Steps:
Apply Power Rule: d/dx (a·xⁿ) = a·n·xⁿ⁻¹
1. Multiply coeff (1) by exp (2) = 2
2. Subtract 1 from exp (2 - 1) = 1
Result: 2x
About this calculator
The Power Rule
The Power Rule is one of the first and most useful rules you learn in calculus. It states that for any real number n, if f(x) = xⁿ, then f'(x) = nxⁿ⁻¹. This allows you to quickly find the slope of polynomial functions without using the limit definition of a derivative.
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Pro Tips
If n = 0, the derivative is 0 (derivative of a constant).
If n = 1, the derivative is just 'a' (derivative of a line).
To find the derivative of a polynomial, apply this rule to each term individually.
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Fun Facts
"The Power Rule was first discovered by Isaac Newton and Gottfried Wilhelm Leibniz independently in the late 17th century."
"It actually works for any real number exponent, including fractions (radicals) and negative numbers (reciprocals), not just integers."
"This simple rule is the foundation for finding rates of change in physics, economics, and engineering."
