Question
Asked Oct 23, 2019
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In this question you will use the definition of a derivative as a limit to find the derivative of f(x) = (x+4)/(x+6)

f(x+h)-f(x)=?

d/dx(x)=

(I got 2/((x+6)^2) for this using the quotient rule

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Expert Answer

Step 1

Given,

(х+4)
f(x)
(х+6)
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(х+4) f(x) (х+6)

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Step 2

Definition of derivative:

f(x+h)-f(x)
f'(x) lim
h
h-0
Now, applying definition of derivative, we get
lim x+h)-f(x)
h 0
f'(x)
(x+h+4) r+4)
(x+h+6) (x+6)
= lim
h 0
(x+h+4)(x+6)-(x+4) (x+h+6)
(x+h+6)(x+6)
lim
h 0
(x+h+4)(x+6)-(x+4)(x+h+6)
lim
h-0
h(x+h+6)x+6)
xh+x2+10x+6h+24- (x2+xh+ 10x+4h+24)
lim
h 0
h(x+h+6) (x+6)
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f(x+h)-f(x) f'(x) lim h h-0 Now, applying definition of derivative, we get lim x+h)-f(x) h 0 f'(x) (x+h+4) r+4) (x+h+6) (x+6) = lim h 0 (x+h+4)(x+6)-(x+4) (x+h+6) (x+h+6)(x+6) lim h 0 (x+h+4)(x+6)-(x+4)(x+h+6) lim h-0 h(x+h+6)x+6) xh+x2+10x+6h+24- (x2+xh+ 10x+4h+24) lim h 0 h(x+h+6) (x+6)

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Step 3

Further calcu...

хh+x2+10х+6h+24-х2—хh-10х-4h-24
= lim
h(x+h+6)(x+6)
h- 0
6h-4h
= lim
h-о h(x+h+6)(x+6)
2h
lim
h-о h (x+h+6) (х+6)
2
lim
h-0 (x+h+6)(x+6)
2
2
(x+0+6)(x+6)
(x+6)2
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хh+x2+10х+6h+24-х2—хh-10х-4h-24 = lim h(x+h+6)(x+6) h- 0 6h-4h = lim h-о h(x+h+6)(x+6) 2h lim h-о h (x+h+6) (х+6) 2 lim h-0 (x+h+6)(x+6) 2 2 (x+0+6)(x+6) (x+6)2

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Tagged in

Math

Calculus

Derivative