Derivatives

# Derivative

The** rate of change** of a function f in an interval [x_{1},x_{2}] is:

*RC = f(x _{2}) – f(x_{1})*

The **average rate of change** of a function f in an interval [x_{1},x_{2}] is

Example: f(x) = x^{2}

- If f is decreasing in the interval, then ARC < 0

- If f is increasing in the interval, then ARC > 0

- If f is constant in the interval, then ARC = 0

Let

*f*defined in an open interval centered at*a*, the**derivative, f’(a), of a function y = f(x) in x = a**is the limit:When the function is continuous and the limit exists, we say that

**f is derivable in x = a.**

As you can see,

*the derivative is the slope of the tangent line at the curve in this point*. It is also called**instantaneous rate of change.**

The equation of the tangent line to the curve at the point is:

Example: if f(x) = x

^{2}+ 1, calculate f’(1) and the tangent line at this point.**Exercises**:

1) Calculate the average rate of change of these functions in the intervals:

2) Calculate the derivative of these functions at these points by using the definition:

Solutions:1) a) -3; b) 1; 2) a) 1; b) 5

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