# Differentiating exponential functions

k guys.. i'd like to try to figure out how to get to the derrivatives on my own so really just looking for a right or wrong.

1. f(x) = e^(3x^2+x) find f'(2)

f'(x) = e^(3x^2+x) * (6x+1)

f'(2)= e^(3(2)^2+2) * [6(2)+1]
f'(2)=13e^14 ... correct?

2) find slope of the tangent to the function f(x) = 2^(x^2+3x) when x=3

f'(x) = (2x+3) 2^(x^2+3x) ln2

f'(3)= [2(3)+3] 2^[3^2+(3)(3)] ln 2
f'(3) = 1 635 339.37 ..... correct?

3) y = xe^(2x)
y' = 2x^2e^(2x) + e^(2x) .... correct?

4) f(x) = 3^x + x^3
f'(x= 3^x ln 3 + 3x^2 .... correct?

5) y= e^-(2x+5)
y' = e^(-2x-5) * (-2) ... correct?

6) y = e^(3x^2-5x+7)
y' = e^(3x^2-5x+7) * (6x-5) ... correct?

If anyone can quickly tell me if these are differentiated correctly it would be very helpful. I'm new to doing this type of work and find myself second guessing most my results.
thanx for any help.

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SammyS
Staff Emeritus
Homework Helper
Gold Member
They look fine.

For #3, you could factor out e2x.

Well ... Dick is right, there is a problem with #3.

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Dick
Homework Helper
I would check (3) again.

thanx guys... did you mean there's a problem with 3 other than i didnt factor out the e^2x?

I've been trying to work 3 differently but keep coming up with the same.. could someone show me where I'm going wrong please?

Dick
Homework Helper
I've been trying to work 3 differently but keep coming up with the same.. could someone show me where I'm going wrong please?
Your first term in 3 should be x time the derivative of e^(2x). What's the derivative of e^(2x)?

the derivative of e^(2x) is... e^(2x)*(2x)..correct?

ohh wait i see.. just 2 not 2x lol

so the final answer should be y'=e^(2x)*(2x+1) ...?

Dick
Homework Helper
ohh wait i see.. just 2 not 2x lol
You got it.

thanx a lot. ... I have another question about graphing.. If I'm given a graph of f'(x).. and it asks me to find a possible graph of f(x) and f''(x)... should I know the shape of f''(x) based off of f'(x) or am I just finding a possible graph for that as well?

Dick
Homework Helper
To make a sketch of f''(x) just think about what the derivative of f'(x) should look like. f''(x) is just the slope of the curve f'(x). Right? To sketch a graph of f(x) think about what the integral of f'(x) would look like.

oh ok. So if I'm just given a graph and representing the function and not the equation of the function I'm just showing what the graphs should look like. Not what they actually look like.
thanx dick... you know, its not true what they say about you... you really dont like up to the name lol

Dick
Homework Helper
oh ok. So if I'm just given a graph and representing the function and not the equation of the function I'm just showing what the graphs should look like. Not what they actually look like.
thanx dick... you know, its not true what they say about you... you really dont like up to the name lol
Well, your sketch of the graph of f''(x) should look pretty close to the graph you would get if you were given an equation. Where f'(x) is increasing, f''(x) better be positive, etc. The sketch of f(x) has a little more variable. Since integration give you a +C arbitrary constant you can shift the graph up or down. You are very welcome. And thanks for the compliment :).

ok... I'm going to add some pics.. could you let me know if there's anything i should change?

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wait the last one is the wrong pic...

this is h''(x)

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Dick
Homework Helper
wait the last one is the wrong pic...
If you were going to multiply the last one by (-1), then everything looks ok. And you can shift the first one by any constant up or down, and it would still be ok.

Dick