Differentiate terms like aX^n -- where X is a variable and both a and n are numbers -- by following a simple pattern: if Y = aX^n then the derivative of Y with respect to X is dY/dX = anX^(n-1). To find the derivative of a polynomial, do one term at a time. For example, if Y = 2X^3 + 5X^2 - 13X + 17, the derivative is dY/dX = 6X^2 + 10X - 13. The 17 disappears because the derivative of a constant is zero. When X changes, the constant does not change -- that is the nature of constants.
Compute the derivative of a product by using the product rule. This is usually described in "function notation." Instead of saying Y = X^2 + 1, calculus uses the notation f(x) = x^2 + 1. In this notation, instead of saying dY/dX = 2X. The derivative is written f '(x) = 2x. Using function notation, the product rule states that when f(x) = g(x) h(x), The derivative of f(x) is given by f '(x) = g(x) h '(x) + h(x) g '(x). For example, if f(x) = x^2(x - 1) we can write f(x) = g(x)h(x) where g(x) =x^2 and h(x) = x - 1. So f'(x) = g(x) h '(x) + h(x) g '(x) = x^2(1) + 2x(X - 1) = 3x^2 - 2x.
Find the derivative of a fraction with the division rule. The derivative of f(x) = g(x) / h(x) is given by f '(x) = ( g '(x)h(x) - g(x) h '(x) ) / h(x)^2. For example if f(x) = (x - 1)/x then g(x) = x - 1 and h(x) = x, so f '(x) = (x(1) - (1)(x - 1))/x^2 = 1/x^2.
Apply the chain rule for the function of a function. The chain rule says that if f(x) = g(h(x)) then f '(x) = g '(h(x))h '(x). For example if f(x) = (2x -1)^3 then g(x) = x^3 and h(x) = 2x - 1, so f '(x) = 3(2x - 1)^2(2) = 6(2x - 1)^2.