3.1 – Derivative of a Function Ch. 3 – Derivatives 3.1 – Derivative of a Function

The derivative of f at x=a can also be found by… The rate of change, or slope, of a function is called its derivative. It is denoted by f’(x), which is read as “f prime of x”. The derivative is an equation for the slope of the tangent line at any point (x, f(x)). If f’(x) exists for some value x, then we say f is differentiable at x. A function differentiable at every point in its domain is a differentiable function. The derivative of f at x=a can also be found by…

Ex: Find the derivative of f(x)=2x2 when x=-1. Method 1: Method 2:

The following symbols indicate the derivative of a function y=f(x) The following symbols indicate the derivative of a function y=f(x). THEY ALL MEAN THE SAME THING! Read as “y prime” “f prime” “dy dx” or “the derivative of y with respect to x” “df dx” “d dx of f at x” or “the derivative of f at x”

Graphing f’ from f Graph the derivative of the function f shown below. Use key points to generate the graph. f(x) + + – + Step 1: Identify zeros (where slope is a horizontal line) Step 2: Identify positive/negative slope ranges between zeros Step 3: Identify how positive/negative slope will be Step 4: Graph the derivative

Alternate Def’n for Differentiability (3.2) If f(x) is continuous at x=a, then f(x) is differentiable at a if… Ex: Is g(x) differentiable over the real numbers? g(x) is definitely differentiable for every value besides zero, so lets check the left and right derivatives at zero. Since the derivatives to the left and right of zero aren’t equal, g(x) is not differentiable at x=0.