
Use Definition to Find First Derivative  Examples with Solutions
Example 1
f(x) = m x + b
Solution to Example 1 We first need to calculate the difference quotient.
\dfrac{f(x+h)f(x)}{h} = \dfrac{m(x+h)+b (mx+b)}{h} \\\\
= \dfrac{mh}{h} = m
The derivative f ' is given by the limit of m (which is a constant) 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
f(x) = a x^2 + bx + c
Solution to Example 2 We first find difference quotient
\dfrac{f(x+h)f(x)}{h} = \dfrac{a(x + h)^2 + b(x + h) + c  ( a x^2 + b x + c )}{h}
Expand the expressions in the numerator and group like terms.
= \dfrac{a x^2 + 2 a x h + a h^2 + b x + b h + c  a x^2  b x  c}{h}
= \dfrac{2 a x h + b h + 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
Solution to Example 3 We first calculate the difference quotient
\dfrac{f(x+h)f(x)}{h} = \dfrac{\sin (x + h)  \sin x }{h}
Use the trigonometric formula to transform a difference sin (x + h)  sin x in the numerator into a product.
\dfrac{\sin (x + h)  \sin x }{h} = \dfrac{2 cos [ (2 x + h)/2 ] sin (h/2)}{h}
Rewrite the above difference quotient as follows.
\dfrac{f(x+h)f(x)}{h} = \dfrac{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
Solution to Example 4 The difference quotient is given by
\dfrac{f(x+h)f(x)}{h} = \dfrac{\sqrt{x+h}  \sqrt x}{h}
Multiply numerator and denominator by √ (x + h) + √ x , expand, group like terms and simplify.
= \dfrac{\sqrt{x+h}  \sqrt x}{h} \times \dfrac{\sqrt{x + h} + \sqrt x}{\sqrt{x + h} + \sqrt x}
= \dfrac{(\sqrt{x+h})^2 (\sqrt x)^2}{h(\sqrt{x + h} + \sqrt x)} = \dfrac{h}{h(\sqrt{x + h} + \sqrt x)} = \dfrac{1}{\sqrt{x + h} + \sqrt x}
As h >0, 1 / [ √ (x + h) + √ x ] > 1 / [ 2 √ (x) ] . Hence the derivative of √ x is 1 / [2 √ x]
f '(x) = \dfrac{1}{2\sqrt x}
Example 5
Solution to Example 5 The difference quotient is given by
\dfrac{f(x+h)f(x)}{h} = \dfrac{1/(x+h)  1/x}{h}
Set the two rational expressions in the numerator to the same denominator and rewrite the above as.
= \dfrac{\dfrac{x}{x(x+h)}  \dfrac{x+h}{x(x+h)}}{h}
= \dfrac{x(x+h)}{x(x+h)h}
Simplify
= \dfrac{1}{x(x+h)}
As h >0,  1 / [x(x+h)] >  1 / x^{2} . Hence the derivative of 1/ x is  1 / x^{2}
f '(x) = \dfrac{1}{x^2}
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