Maths Exponentials Help

[acroviak]

Hey
Im stuck with a few questions i have been set.

Firstly, I need to work out the integration of
y = e^x
y = e^x/2
y=e^0.5x and
y = e^2x

Secondly, i need to differentiate
1.) y=ln x^7 / 7
2.) y= e^2x
3. y = 4e^x


and lastly, i need to solve a differential equation.

dy/dx = 1/x + x^2 (given that y=0 when x=1)



thanks for all the help.

i managed to do it all apart from them few questins

[acroviak]

also knowing the integration of y=e^0.5x would help

[Magyar]

You might be better off posting this question in the Science section.

[SpodoCommodo]

d( e^u )/dx = (e^u) * (du/dx)

d( ln u ) = (1/u) * (du/dx)

The questions you give are all straightforward examples of these 2 general equations e.g. the integral of 1/x = ln x

[factor-fiction]

Yes, on the first 4, take logs of both sides, then differentiate both sides. For example, the first one becomes
logy= xloge= x
differentiating both sides:
(1/y)dy/dx=1
dy/dx=y= e^x

[SpodoCommodo]

There is definitely no need to take logs of anything.

Lets use an example, u = 2x.

therefore du/dx = 2

so d(e^u)/dx = (e^u) * 2

i.e. d (e^2x)/dx = 2*e^2x

That's one of your questions done explicitly. But the whole point is to learn the method not just be given the answers.

[factor-fiction]

I misread the question and differentiated rather than integrated the first one- but still got the right answer because it's e^x in both cases.

Both methods are fine. Taking logs can be useful.

How would you differentiate x^x?. I'd take logs then- is there an easier way?

[SpodoCommodo]

For x^x then taking logs is a good idea but in these simple examples it's an unnecessary extra step (although it inevitably gives the right answer).