logarithmic differentiation: 4lnx + ((30-x)/10)^2

[res publica]

Hello everybody, I need to find the derivative of the function 4lnx+((30-x)/10)² and set it equal to zero, but somehow I feel unable to do so. Could anybody please help me to find the derivative and explain me exactly how he got it?

A second but similar problem: y= e^-x²

Sketch: http://www.imagehack.eu/de/uploads/6685be7bdb.jpg

I need to find the coordinates of C when the rectangle under the curve e^-x² and on the x-axis is as big as possible. How can I solve this problem?

Thank you for your time and effort.

 

[galactus]

For the second one, you have \(\displaystyle \L\\y=e^{-x^{2}}\)

The area of the rectangle is A=2xy.

Sub in y, differentiate, set to 0 and solve for x. That will be the x-coordinate of the largest possible rectangle. y will easily follow by subbing back in.

 

[arthur ohlsten]

y=4 ln x + ((30-x)/10)^2 take derivative with respect to x
dy/dx = d/dx[4 lnx ] + d/dx[ [ 30-x]/10]^2
dy/dx= 4 d/dx lnx + d/dx [ [30-x^2]/100 ]

but d/dx ln x = 1/x
dy/dx =4 /x + [1/100] d/dx [30-x]^2

let u= [30-x] then
du=d/dx[30] + d/dx[-x]
du=-dx
du/dx=-1

dy/dx =4/x +[1/100] d/dx u^2
dy/dx =4/x +[1/100]2 u du/dx
dy/dx=4/x +[1/50] [30-x][-1]

part 2 to follow

Arthur
dy/dx =4/x + [x-30]/50

 

[res publica]

I could do it, thanks a lot for your help.

 

[arthur ohlsten]

your welcome
Arthur