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Differentiating a Logarithmic Function In Exercise, find the derivative of the function.

f(x) = In x^2 (x + 1)^1/2

User Karlingen
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1 Answer

3 votes

Answer:


(dy)/(dx)=(4x+2)/(x^2+x)

Explanation:

Data provided in the question:


y =\ln(x^(2) (x+1)^(1)/(2))

now,

we know

ln(AB) = ln(A) + ln(B)

Therefore,


y =\ln(x^(2)) + \ln((x+1)^(1)/(2))

also,

ln(aᵇ) = b × ln(a)

thus,

y =
2\ln(x) + (1)/(2)*\ln(x+1)

differentiating with respect to 'x' , we get


(dy)/(dx)=2*(1)/(x)+((1)/(2))((1)/(x+1))*(d(x+1))/(dx)

[∵ derivative of ln(a) =
(1)/(a) *(d(a))/(dx)) ]

or


(dy)/(dx)=(2)/(x)+((1)/(2))((1)/(x+1))*1

or


(dy)/(dx)=(2)/(x)+(1)/(2(x+1))

or


(dy)/(dx)=(2*2(2x+1))/(x*2(x+1))

or


(dy)/(dx)=(4x+2)/(x^2+x)

User Adelia
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