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 [x0, x1] and its limit as x1 x0 is the instantaneous rate of change of f(x) at x=x0.
Derivative of a function
Suppose that x0 is a number in the domain of a function f then the derivative of ‘f’ at x=x0 and is denoted by f’(x0) and is defined as
If the limit of the difference quotient exists, f’(x0) is the slope of the graph of ‘f’ at the point x=x0. If this limit does not exist, then the slope of the graph of ‘f’ is undefined at x=x0
Example: Find the slope of the function y=x2+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 x0 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(x0, f(x0) to be the line whose equation is y – f(x0) = f’(x0)(x-x0).
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
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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