Derivative

Derivative

Objectives: In this tutorial, using the definition of derivative, we derive a number of elementary differentiation formulas. Some examples are provided to show the use of these formulas. After working through these materials, the student should be able

to derive symbolically some of the elementary differentiation formulas; and
to calculate symbolically the derivative of some functions using these formulas.

Modules:

Definition. Let y = f(x) be a function. The derivative of f is the function whose value at x is the limit

provided this limit exists.

Theorem A. The derivative of a constant is 0.

Theorem B. The derivative of the identity function f(x) = x is the constant function f '(x) = 1.

Theorem C. If the function g(x) = k f(x) where k is a constant and f is a differentiable function, then g '(x) = k f '(x).

Theorem D. If f and g are differentiable functions, then (f + g)'(x) = f '(x) + g '(x)

Corollary. If f and g are differentiable functions, then (f - g)'(x) = f '(x) - g '(x)

Theorem E. (Power Rule) If f(x) = xⁿ where n is a positive integer, then f '(x) = n x^n - 1

Example 1: f(x) = 3x³ + 6x² + 2x + 8

Example 2: f(x) = 8x³ + x² + 4x + 3

Example 3: f(x) = x³ + 11x² + 5x + 9

Example 4: f(x) = 7x³ + 3x² + x + 3