The basic derivative rules in calculus are formulas that help you find the rate of change of functions quickly without using limits every time. These rules include the power rule, constant rule, constant multiple rule, sum and difference rules, product rule, quotient rule, and chain rule.
Steps
The constant rule states that the derivative of any constant is zero, since constants don't change.
The power rule says to multiply by the exponent and reduce the exponent by one: the derivative of x^n is n·x^(n-1).
The constant multiple rule allows you to pull constants out: the derivative of c·f(x) is c·f'(x).
The sum and difference rules let you take derivatives term by term: the derivative of f(x) + g(x) is f'(x) + g'(x).
The product rule handles multiplication of functions: the derivative of f(x)·g(x) is f'(x)·g(x) + f(x)·g'(x).
The quotient rule handles division: the derivative of f(x)/g(x) is [f'(x)·g(x) - f(x)·g'(x)] / [g(x)]^2.
The chain rule tackles composite functions: the derivative of f(g(x)) is f'(g(x))·g'(x), meaning you take the derivative of the outer function and multiply by the derivative of the inner function.
Worked example
Find the derivative of f(x) = 3x^4 - 2x + 5. Using the constant multiple rule and power rule on the first term: derivative of 3x^4 is 3·4·x^3 = 12x^3. Using the power rule on the second term: derivative of -2x is -2. Using the constant rule on the third term: derivative of 5 is 0. Combining these using the sum rule gives f'(x) = 12x^3 - 2.
Remember
Mastering these basic derivative rules allows you to find rates of change for most functions you'll encounter without returning to the limit definition each time.