Power Rule for Derivatives
If f(x) = x^n, then f'(x) = n*x^(n-1)
The power rule is a fundamental rule in calculus that allows us to take the derivative of a function raised to a power. The power rule states that for any positive integer n, if y = x^n, then dy/dx = nx^(n-1).
In simpler terms, if we have a function f(x) = x^n, we can find its derivative f'(x) as follows:
f'(x) = (d/dx)(x^n)
= nx^(n-1)
For example, if we have a function f(x) = x^3, we can find its derivative as follows:
f'(x) = (d/dx)(x^3)
= 3x^(3-1)
= 3x^2
So, the derivative of f(x) = x^3 is f'(x) = 3x^2.
The power rule also holds for functions that have negative exponents or fractional exponents. For example, if we have a function f(x) = x^(-2), we can find its derivative as follows:
f'(x) = (d/dx)(x^(-2))
= -2x^(-2-1)
= -2x^(-3)
So, the derivative of f(x) = x^(-2) is f'(x) = -2x^(-3).
In summary, the power rule is a powerful tool in calculus that allows us to find the derivatives of functions raised to any power.
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