Find the derivative of y = 5x + 10)^(x)

[carson_6969]

1. y=5x+10)^(x)


x(5x+10)^(-1)5
5x(5x+10)
25x^(2) + 50x
25x(x+2)

 

[arthur ohlsten]

let me assume the equation is:
y=[5x+10]^x take ln
ln y = x ln [5x+10] take derivative
1/y dy/dx = x[5/[5x+10]] + ln [5x+10]
dy/dx = [5x+10]^x + ln [5x+10] + 5x/[5x+10] answer
dy/dx = [5^x][x+2]^x + ln 5 + ln [x+2] + x/[x+2] answer

please check math for errors
Arthur

 

[morson]

Could be written as \(\displaystyle \L\ 5^x(x + 2)^x\), but arthur's way seems easier.

\(\displaystyle \L\ ln(y) = xln(5(x + 2))\)

\(\displaystyle \L\ ln(y) = x ln(5) + x ln(x + 2)\)

\(\displaystyle \L\ \frac{1}{y}\ \frac{dy}{dx}\ = ln(5) + ln(x + 2) + \frac{x}{x + 2}\\)

So, \(\displaystyle \L\ \frac{dy}{dx}\ = (5x + 10)^x [ ln(5) + ln(x + 2) + \frac{x}{x + 2}\]\)

 

[arthur ohlsten]

I always suggest the student uses whatever method is easiest for him.
"There are many ways to skin a cat"

Now that you know the answer I suggest you redo the problem in whatever technique appears to come naturally
Arthur

 

[wjm11]

A minor point:

Arthur said:
1/y dy/dx = x[5/[5x+10]] + ln [5x+10]
dy/dx = [5x+10]^x + ln [5x+10] + 5x/[5x+10] answer

I think there may be some parentheses missing(?) Second line should be
dy/dx = ([5x+10]^x)( ln [5x+10] + 5x/[5x+10])

 

[arthur ohlsten]

you are right there is a parenthesis missing
y=[5x+10[^x take ln
lny=x ln[5x+10] take derivative
1/y dy/dx= x[5/[5x+10]] + ln[5x+10]
dy/dx = [5x+10]^x { x[5/[5x+10]]+ln[5x+10]} answer
dy/dx=[5x+10]^x { x/[x+2] + ln5 + ln [x+2] ) answer
Arthur