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

Answer:

a) $f'(x)=6$

b) $f'(x)=12$

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

Step-by-step explanation:

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

Solution :

Definition of the derivative is

$f'(x)= \lim_{h \to 0}(\frac{f(x+h)-f(x)}{h})$

Applying in the functions,

a) $f(x)=6x$

$f'(x)= \lim_{h \to 0}(\frac{6(x+h)-6x}{h})$

$f'(x)= \lim_{h \to 0}(\frac{6x+6h-6x}{h})$

$f'(x)= \lim_{h \to 0}(\frac{6h}{h})$

$f'(x)=6$

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

$f'(x)= \lim_{h \to 0}(\frac{12(x+h)-2-(12x-2)}{h})$

$f'(x)= \lim_{h \to 0}(\frac{12x+12h-2-12x+2}{h})$

$f'(x)= \lim_{h \to 0}(\frac{12h}{h})$

$f'(x)=12$

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

$f'(x)= \lim_{h \to 0}(\frac{k(x+h)^2-kx^2}{h})$

$f'(x)= \lim_{h \to 0}(\frac{k(x^2+h^2+2xh-kx^2)}{h})$

$f'(x)= \lim_{h \to 0}(\frac{kx^2+kh^2+2kxh-kx^2}{h})$

$f'(x)= \lim_{h \to 0}(\frac{h(kh+2kx)}{h})$

$f'(x)= \lim_{h \to 0}(kh+2kx)$

$f'(x)=2kx$