Use Definiton to Find Derivative
Use the definition of the derivative to differentiate functions. This tutorial is well understood if used with the
difference quotient
and the
definition of the derivative.
The derivative f ' of function f is defined as
f '(x) = lim_{ h >0} [f(x+h)  f(x)] / h
when this limit exists. Hence, to find the derivative from its definition, we need to find the limit of the difference quotient as h approaches zero.
Example 1: Use the definition of the derivative to find the derivative of function f defined by
f(x) = m x + b
Solution to Example 1

We first need to calculate the difference quotient.
[ f (x + h) – f(x) ] / h = [ m (x + h) + b – (m x + b) ] / h
= [ m h ] / h = m

The derivetive f ' is given by the limit of m as h >0. Hence
f '(x) = m

The derivative of a linear function f(x) = m x + b is equal to the slope m of its graph.
Example 2: Use the definition to find the derivative of
f(x) = ax^{ 2} + bx + c
Solution to Example 2

We first find difference quotient
[ f (x + h) – f(x) ] / h =
[ a(x + h)^{ 2} + b(x + h) + c  ( ax^{ 2} + bx + c ) ] / h

Expand the expressions in the numerator and group like terms.
= [ ax^{ 2} + 2 a x h + a h^{ 2} + bx + bh + c  ax^{ 2}  bx  c] / h
= [ 2 a x h + bh + a h^{ 2}] / h = 2 a x + b + a h

The limit of 2 a x + b + a h as h >0 is equal to 2 a x + b. Hence
f '(x) = 2 a x + b
Example 3: Find the derivative, using the definition, of function f given by
f(x) = sin x
Solution to Example 3

We first calculate the difference quotient
[ f (x + h) – f(x) ] / h =
[ sin (x + h)  sin x ] / h

Use the trigonometric formula to transform a difference sin (x + h)  sin x into a product.
[ sin (x + h)  sin x ] / h = 2 cos [ (2 x + h)/2 ] sin (h/2) / h

Rewrite the above difference quotient as follows.
[ f (x + h) – f(x) ] / h = cos [ (2 x + h)/2 ] sin (h/2) / [ h / 2 ]

As h >0, sin (h/2) / [ h / 2 ]>1 and cos [ (2 x + h)/2 ] > cos (2x /2) = cos x. Hence the derivative of sin x is cos x
f '(x) = cos x
Example 4: Use the deinition to differentiate
f(x) = sqrt x
Solution to Example 4

The difference quotient is given by
[ f (x + h) – f(x) ] / h =
[ sqrt (x + h)  sqrt x ] / h

Mutliply numerator and denominator by [ sqrt (x + h) + sqrt x ], expand, group like terms and simplify.
= [ sqrt (x + h)  sqrt x ][ sqrt (x + h) + sqrt x ] / [ h ( sqrt (x + h) + sqrt x ) ]
= h / [ h ( sqrt (x + h) + sqrt x ) ]
= 1 / [ sqrt (x + h) + sqrt x ]

As h >0, 1 / [ sqrt (x + h) + sqrt x ] > 1 / [ 2 sqrt (x) ] . Hence the derivative of sqrt x is 1 / [2 sqrt x]
f '(x) = 1 / [2 sqrt x]
More on differentiation and derivatives  
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Updated: 2 April 2013
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