Power Rule
In calculus, the power rule is a fundamental rule used to differentiate a function that is in the form of a power of x
In calculus, the power rule is a fundamental rule used to differentiate a function that is in the form of a power of x. It provides a way to find the derivative of a function by simply multiplying the original power by the coefficient in front of x and then reducing the power by 1.
The power rule states that for any real number n and a function f(x) = x^n, the derivative of f(x) is given by:
f'(x) = nx^(n-1)
Let’s go through a few examples to better understand how to use the power rule:
1. Differentiate f(x) = x^3:
Using the power rule, we take the original power of x (3) and multiply it by the coefficient in front of x (which is 1 in this case). Then we reduce the power by 1.
f'(x) = 3x^(3-1) = 3x^2
2. Differentiate g(x) = 5x^2:
Again, we apply the power rule by multiplying the original power of x (2) by the coefficient (which is 5). Then we reduce the power by 1.
g'(x) = 2 * 5x^(2-1) = 10x
3. Differentiate h(x) = 4x^4:
Applying the power rule, we obtain:
h'(x) = 4 * 4x^(4-1) = 16x^3
The power rule is a powerful tool in calculus as it allows us to easily find the derivative of functions that are in the form of powers of x. It is based on the observation that the slope of a tangent line to a curve is proportional to the exponent of x in the equation. Knowing the power rule is crucial in many areas of mathematics and science where differentiation is required to analyze rates of change.
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