Rules for Derivatives

Finding derivatives becomes much easier when you know the basic rules. Let's break them down:

Rule 1: Derivative of a constant is always zero. If f$x$ = 3, then f'$x$ = 0. Constants don't change as x changes, so their rate of change is zero!

Rule 2: Power Rule states that if f$x$ = ax^n, then f'$x$ = n·ax^$n-1$. This rule is super useful - you multiply by the power and then reduce the power by 1. For example, the derivative of x^4 is 4x^3.

Rule 3: Constant Multiple Rule lets you pull constants outside the derivative. For f$x$ = 3x², f'$x$ = 3(2x) = 6x.

Rule 4: Sum Rule means you can find derivatives term by term. For f$x$ = x⁴ + 12x², f'$x$ = 4x³ + 24x.

Rule 5: Product Rule helps when multiplying functions. If f$x$ = u·v, then f'$x$ = (u·dv) + (v·du). For example, with f$x$ = 2x-5$$3x+2, f'$x$ = $2x-5$(3) + $3x+2$(2).

Remember This! The Product Rule follows the pattern "first times derivative of second, plus second times derivative of first."

Rule 6: Quotient Rule works for fractions. If f$x$ = u/v, then f'$x$ = (v·du - u·dv)/v². Think "low d-high minus high d-low, over low squared."

Rule 7: Power Rule for Negative Exponents works just like the regular power rule. For f$x$ = 2x^$-3$, f'$x$ = -6x^$-4$.

The exponential function has a special property: the derivative of e^x is e^x - it's its own derivative!

Second derivatives (f"$x$) are just the derivatives of the first derivatives. They measure how the rate of change is itself changing.