1. Find the derivative using the definition of derivative as a limit. a) f(x)= -
ID: 2873103 • Letter: 1
Question
1. Find the derivative using the definition of derivative as a limit.a) f(x)= -5x b) f(x)= 1 / x-1 c(x)= sqrt foot x+1
2. Determine whether the function is differentiable at x=2
F(x)= { 1/2x+1 x<2 { sqrt 2x x> or = 2 1. Find the derivative using the definition of derivative as a limit.
a) f(x)= -5x b) f(x)= 1 / x-1 c(x)= sqrt foot x+1
2. Determine whether the function is differentiable at x=2
F(x)= { 1/2x+1 x<2 { sqrt 2x x> or = 2 1. Find the derivative using the definition of derivative as a limit.
a) f(x)= -5x b) f(x)= 1 / x-1 c(x)= sqrt foot x+1
2. Determine whether the function is differentiable at x=2
F(x)= { 1/2x+1 x<2 { sqrt 2x x> or = 2
Explanation / Answer
Solution:(1)
(a) f(x) = -5x
f '(x) = lim(h0) (f(x+h) - f(x))/h
= lim(h0) (-5(x+h) - (-5x))/h
= lim(h0) (-5x-5h+5x))/h
= -5
(b)
f(x) = 1/(x-1)
= lim(h0) (f(x+h) - f(x))/h
= lim(h0) (1/(x+h-1) - 1/(x-1))/h
= lim(h0) [{(x-1) -(x+h-1)}/(x+h-1)(x-1)]/h
= lim(h0) [-h/(x+h-1)(x-1)]/h
= lim(h0) [-1/(x+h-1)(x-1)]
= -1/(x-1)^2
(c)
f(x) = Sqrt(x+1)
= lim(h0) (f(x+h) - f(x))/h
= lim(h0) (sqrt(x+h+1) - sqrt(x+1))/h
= lim(h0) [{sqrt(x+h+1) - sqrt(x+1)}{sqrt(x+h+1) + sqrt(x+1)}/{sqrt(x+h+1) + sqrt(x+1)}]*(1/h)
= lim(h0) [{(x+h+1) - (x+1)}/{sqrt(x+h+1) + sqrt(x+1)}]*(1/h)
= lim(h0) [{h}/{sqrt(x+h+1) + sqrt(x+1)}]*(1/h)
= lim(h0) [1/{sqrt(x+h+1) + sqrt(x+1)}]
=1/ x+1
Solution; (2)
As x2 from the left:
f'(x) = 1/2
f'(2-) = 1/2
As x2 from the right
f'(x) = 1/(2x)
f'(2+) = 1/(2*2) = 1/2
f'(2-) = f'(1+)
The equation is differentiable at x=2
f'(2) = 1/2
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