Logarithmic Function

2015-06-07 1:51 am

1. Let y=√[(x-2)(3x+4)/(5x-6)] , for x>2. Find dy/dx by logarithmic differentiation.

2. Find dy/dx of each of the following functions.
(a) y=x^(sinx) , x>0
(b) y=x^(4x) , x>0
(c) y=5^x
(d) y=(cosx)^x , for cosx>0

3. FInd the derivative of the function y=x^(tanx) with respect to x at x=(π/4).

回答 (2)

pui

2015-06-07 3:57 am

✔ 最佳答案

1.
y=√[(x-2)(3x+4)/(5x-6)]
y^2=(x-2)(3x+4)/(5x+6)
2In y=In (x-2)+In(3x+4) -In(5x+6)
(2/y)(dy/dx)=1/(x-2) +3/(3x+4) -5/(5x+6)
dy/dx=(y/2)[1/(x-2) +3/(3x+4) -5/(5x+6)]

2a
y=x^(sinx)
In y=(sin x)(In x)
(1/y)(dy/dx)=(cos x)(In x)+(sin x)/x
dy/dx=y[(cos x)(In x)+(sin x)/x]

2b
y=x^(4x)
(In y)(dy/dx)=4x In x
(1/y)(dy/dx)=(4x/x)+4 Inx
dy/dx=4y(1+In x)

2c
y=5^x
dy/dx=(5^x)(In x)

2d
y=(cosx)^x
In y =x(In cos x)
(1/y)(dy/dx)=-(x sinx)/(cosx)+In cosx
dy/dx=y(-x tanx +In cosx)

3
y=x^(tan x)
In y=(tan x)(In x)
(1/y)(dy/dx)=(In x)/[(cosx)(cosx)] +(tan x)/x
dy/dx=y{In x /[(cosx)(cosx)] +(tanx)/x}
dy/dx|x=(π/4) =0.6206

2015-06-09 19:09:33 補充：
sorry
2c
y=5^x
dy/dx=(5^x)(In 5)

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polarbearhmh

2015-06-08 1:35 am

2c
y=5^x
dy/dx=(5^x)(In 5)

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