# Introduction

The main idea of derivative is that of rate of change of a function. The primary mathematical tool that is used to calculate rate of change and the slopes of curves is derivative. Slope of the graph of y=f(x) at x=x0is given by

The ratio is called a difference quotient.

The difference quotient can also be interpreted as the average rate of change of f(x) over the interval [x_{0}, x_{1}] and its limit as x_{1} x_{0} is the instantaneous rate of change of f(x) at x=x_{0}.

## Derivative of a function

Suppose that x_{0} is a number in the domain of a function f then the derivative of ‘f’ at x=x_{0} and is denoted by f’(x_{0}) and is defined as

If the limit of the difference quotient exists, f’(x_{0}) is the slope of the graph of ‘f’ at the point x=x_{0}. If this limit does not exist, then the slope of the graph of ‘f’ is undefined at x=x_{0}

Example: Find the slope of the function y=x^{2}+1 at the point (2, 5)

Solution: Slope of the curve at the point (2, 5) is given by

## Graphical representation of Derivative

Suppose that x_{0} is a number in the domain of a function ‘f’. If

then we define the tangent line to the graph of f at the point P(x_{0}, f(x_{0}) to be the line whose equation is y – f(x_{0}) = f’(x_{0})(x-x_{0}).

Below is the graph of a function f(x) and a rough sketch of f’(x). The graph of f’(x) represents the slopes of the tangent lines to a point on the graph for each x-value in the domain of f(x). For example if one draws a tangent line through the point (0.5, 3) its slope will be zero. Looking at the graph of f’(x) at x=0.5, y=0

Example: Find the derivative with respect to x of f(x)=x3-x. Graph f and f’ together and discuss the relationship between the two graphs

Solution:

Since f’(x) can be interpreted as the slope of the graph of y=f(x) at x, the derivative f’(x) is positive where the graph of f has positive slope, it is negative where the graph of f has negative slope, and it is zero where the graph of f is horizontal.

## Interpretation of the derivative

The derivative f’ of a function f can be interpreted as a function whose value at x is the slope of the graph of y=f(x) at x, or alternatively, it can be interpreted as a function whose value at x is the instantaneous rate of change of y with respect to x at x. In particular, when y=f(t) describes the position at time t of an object moving along a straight line, then f’(t) describes the position at time t of an object moving along a straight line, then f’(t) describes the instantaneous velocity of the object at time ‘t’.

From the figure at each value of ‘x’, the tangent line to a line y=mx+b coincides with the line itself and hence all tangent lines have slope m. This suggests geometrically that if f(x) = mx + b, then f’(x) = m for all x. This is confirmed by the following computations:

Example: The graph of y=|x| has a corner at x=0 which implies that f(x)=|x| is not differentiable at x=0

a) Prove that f(x)=|x| is not differentiable at x=0 by showing that the limit does not exist at x=0

b) Find a formula for f’(x)

Solution: From the definition

does not exist because the one sided limits are not equal.

b) A formula for the derivatives of f(x) =|x| can be obtained by writing |x| in piecewise form and treating the cases x>0 and x<0 separately. If x>0, then f(x) =x and f’(x)=1; if x<0, then f(x)-x and f’(x)=-1. Thus,

The graph of f’ is shown below. We can see that f’ is not continuous at x=0. This shows that a function that is continuous everywhere may have a derivative that fails to be continuous everywhere

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### Reference Links:

- http://en.wikipedia.org/wiki/Derivative
- http://en.wikipedia.org/wiki/Difference_quotient
- http://en.wikipedia.org/wiki/Slope
- http://en.wikipedia.org/wiki/Tangent_lines_to_circles