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spayn [35]
3 years ago
14

2. From the definition of the derivative (definition 5.3), find the derivative for each of the following functions: (a) f(x) = 6

x (b) f(x) = 12x – 2 (c) f(x) = kx² for k a constant
Mathematics
1 answer:
castortr0y [4]3 years ago
6 0

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

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