13 Definition of derivative

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1 3 Definition of derivative 3 Definition and geometrical interpretation Te goal ere is to express te slope of te line tangent to te grap of a function In te next section, we will see tat tis slope gives te (instantaneous) rate of cange of te function Pictured below is te grap of a function f We would like to find te slope of te line tangent to te grap of f at te point P (red line) However, in order to find te slope of a line, we need two points on te line (so tat we can take te difference of te y-coordinates over te difference of te x-coordinates) We could use P as one of te points, but tere is no obvious way to come up wit a second point If we pus to te side a small distance, we get a second point Q on te grap Te line troug P and Q is called a secant line (green line) We can find te slope of te secant line by using te two points P and Q: slope of secant = f(x + ) f(x) We can tink of te secant line as being an approximation to te tangent line Tis approximation becomes better and better te closer Q is to P, tat is, te smaller is Terefore, te slope of te tangent line, denoted f (x), is te limit of te slope of te secant line as approaces 0: f (x) f(x + ) f(x) slope of slide Q slope of tangent toward P secant

2 3 DEFINITION OF DERIVATIVE 2 Since x can be any number (for wic te limit exists), te formula above defines a function f, called te derivative of f Definition of derivative Te derivative of te function f is te function f defined by f f(x + ) f(x) (x) An x for wic te above limit does not exist is not in te domain of f Te derivative f is regarded as a general slope function It can be used to find te slope of any line tangent to te grap of f: If P is a point on te grap, ten te slope of te tangent line at P is obtained by evaluating te derivative f at te x-coordinate of P 32 Finding derivative directly from definition Te student wo as ad some calculus before migt know some rules for finding a derivative tat allow one to avoid evaluating a limit We will eventually obtain tese rules However, for te time being we will be finding te derivative of a function f by using te formula f f(x + ) f(x) (x) Tis is called finding te derivative directly from te definition 32 Example Find te derivative of f(x) = x 2 directly from te definition, use it to find te slopes of te lines tangent to te curve at te points wit x-coordinates x = 2,, 0,, 2, and sketc te grap of f togeter wit tese tangent lines We ave f f(x + ) f(x) (x + ) 2 x 2 (x) x 2 + 2x + 2 x 2 2x + = 2x (2x + )

3 3 DEFINITION OF DERIVATIVE 3 Te derivative f (x) = 2x is a general slope function Te slopes of te tangent lines are obtained by evaluating te derivative at te given values of x (see table) Te grap is sown wit te tangent lines tagged wit teir slopes f (x) x Example Find te derivative of f(x) = x + 2 directly from te definition, use it to find an equation of te line tangent to te grap of f at te point P (, ), and sketc te grap of f togeter wit tis tangent line We ave (using te rationalization metod in te process) f f(x + ) f(x) (x + ) + 2 x + 2 (x) (x + ) + 2 x + 2 (x + ) x + 2 (x + ) x + 2 ((x + ) + 2) (x + 2) ( (x + ) x + 2) ( (x + ) x + 2) (x + ) x + 2 = 2 x + 2 Te slope of te line tangent to te grap of f at P (, ) is m = f ( ) = 2, so tis tangent line as equation y = 2 (x ( )), wic as slope-intercept form y = 2 x Te sketc is

4 3 DEFINITION OF DERIVATIVE Example Find te derivative of f(x) = x + x definition directly from te We ave (using te combining fractions metod in te process) f f(x + ) f(x) (x) (x + ) + (x + ) x + x (x + + )(x ) (x + )(x + ) (x + )(x ) (x + )(x ) (x + + )(x ) (x + )(x + ) (x + )(x ) (x 2 x + x + x ) (x 2 + x + x + x ) (x + )(x ) = 2 (x ) 2 2 (x + )(x ) 2 (x + )(x ) Te act of finding te derivative of a function is called differentiation For instance, instead of saying Find te derivative of te function f(x) = x 2, one could say Differentiate te function f(x) = x 2 3 Exercises

5 3 DEFINITION OF DERIVATIVE 5 3 Find te derivative of f(x) = x 2 + 2x directly from te definition, use it to find te slopes of te lines tangent to te curve at te points wit x-coordinates x =, 0,, 2, 3, and sketc te grap of f togeter wit tese tangent lines 3 2 Let f(x) = x 2 (a) Find te derivative of f directly from te definition (b) Find te x-intercept of te line tangent to te grap of f at te point (, ) 3 3 Find te derivative of f(x) = x directly from te definition