Basic Derivative Rules

Ever wondered how to find the rate of change of any function? That's exactly what derivatives help us calculate! The most fundamental derivative rules are your toolkit for solving problems.

The constant rule states that the derivative of any constant is zero. For example, if f(x) = 7, then f'(x) = 0. This makes sense because constants don't change as x changes.

The power rule is your go-to formula for most functions: when f(x) = xⁿ, the derivative is f'(x) = nxⁿ⁻¹. This works for all kinds of powers:

• For x², the derivative is 2x
• For x^(1/2) (or √x), the derivative is (1/2)x^(-1/2) or 1/(2√x)
• For x^(-2) $or 1/x²$, the derivative is -2x^(-3) or -2/x³

Remember this! The sum/difference rule tells us that the derivative of a sum equals the sum of the derivatives. This is why f(x) = 5x² - 3x + 7 has the derivative f'(x) = 10x - 3.

When working with derivatives, keep in mind that differentiability implies continuity, but continuity doesn't guarantee differentiability. Also, derivatives at endpoints or sharp turns are always undefined.