Question

2. From the definition of the derivative (definition 5.3), find the derivative for each of the following functions: (a) f(x) = 6x (b) f(x) = 12x – 2 (c) f(x) = kx² for k a constant

Answer 1

a)
`f'(x)=6`

b)
`f'(x)=12`

c)
`f'(x)=2kx`

Explanation:

To find : From the definition of the derivative find the derivative for each of the following functions ?

Solution :

Definition of the derivative is

`f'(x)= \lim_(h \to 0)((f(x+h)-f(x))/(h))`

Applying in the functions,

a)
`f(x)=6x`

`f'(x)= \lim_(h \to 0)((6(x+h)-6x)/(h))`

`f'(x)= \lim_(h \to 0)((6x+6h-6x)/(h))`

`f'(x)= \lim_(h \to 0)((6h)/(h))`

`f'(x)=6`

b)
`f(x)=12x-2`

`f'(x)= \lim_(h \to 0)((12(x+h)-2-(12x-2))/(h))`

`f'(x)= \lim_(h \to 0)((12x+12h-2-12x+2)/(h))`

`f'(x)= \lim_(h \to 0)((12h)/(h))`

`f'(x)=12`

c)
`f(x)=kx^2` for k a constant

`f'(x)= \lim_(h \to 0)((k(x+h)^2-kx^2)/(h))`

`f'(x)= \lim_(h \to 0)((k(x^2+h^2+2xh-kx^2))/(h))`

`f'(x)= \lim_(h \to 0)((kx^2+kh^2+2kxh-kx^2)/(h))`

`f'(x)= \lim_(h \to 0)((h(kh+2kx))/(h))`

`f'(x)= \lim_(h \to 0)(kh+2kx)`

`f'(x)=2kx`