Finding a derivative using the "Limit Definition" or "Difference Quotient" is tedious and time-consuming. Fortunately, for most algebraic functions, we can use a shortcut known as The Power Rule. This rule allows us to differentiate functions almost instantly.
1. The Formula
The Power Rule applies to any term where x is raised to a constant power n.
[Image of power rule formula graphic with arrows showing the exponent moving to the front]In simple English: "Bring the power down to the front, then subtract one from the exponent."
2. Basic Examples
Let's look at how this works with simple positive integers:
Bring down the 5, subtract 1 from 5.
f'(x) = 5x⁴
2. f(x) = x³
f'(x) = 3x²
3. f(x) = x
Remember x is x¹.
f'(x) = 1x⁰ = 1 (Since anything to power 0 is 1).
3. Handling Coefficients
If there is already a number in front of the x (a coefficient), you simply multiply the old number by the power you bring down.
Bring down the 4 and multiply by 3.
f'(x) = (3 · 4)x³
f'(x) = 12x³
4. The Invisible Powers (Roots and Fractions)
The Power Rule is incredibly versatile because it also works for negative numbers (fractions) and fractions (roots/radicals). You just have to rewrite the term first.
[Image of graph showing y equals square root of x and its tangent line]A. Negative Exponents (1/x)
If you have 1/x², rewrite it as x⁻².
Bring down -3, subtract 1 (be careful: -3 - 1 = -4).
f'(x) = -3x⁻⁴
Rewrite: -3 / x⁴
B. Fractional Exponents (Roots)
If you have √x, rewrite it as x¹/².
Bring down 1/2, subtract 1 (1/2 - 1 = -1/2).
f'(x) = (1/2)x⁻¹/²
Rewrite: 1 / (2√x)
5. Derivative of a Constant
What is the derivative of a plain number, like 5? A constant line is horizontal. A horizontal line has a slope of 0. Therefore, the derivative of any constant is zero.
f'(x) = 0