# Tangent Lines and Derivatives

The following phrases all mean the same thing:

• the slope of f at a
• the derivative of f at a
•  f ' (a)
• the slope of the tangent line to f at a
• the instantaneous rate of change of f at a

Since f ' (a) (the derivative of f at a) is equal to the slope of the tangent line to f at a, we can determine the sign of f ' (a) by looking at the tangent line to f at a.

### Sample Problem

Consider this function:

The tangent line to f at a is sloping upwards. This means the slope of the tangent line to f at a is positive. Since the slope of the tangent line to f at a equals f ' (a), we know f ' (a) is also positive.

We can also tell the sign of f ' (a) by looking at the function f and not thinking about tangent lines.

• If f is increasing at a then f ' (a) is positive.

• If f is decreasing at a then f ' (a) is negative.

• If f "flattens out" at a then f ' (a) is zero.

We can also use tangent lines to compare values of f ' at different points.

Remember these pictures. We'll use them again when we discuss concavity and second derivatives.