# Derivatives of Exponential Functions

• Aug 24th 2011, 11:02 AM
raymac62
Derivatives of Exponential Functions
Hi,

I was wondering if someone could check my work for the following two short problems:

1. If $f(x)=e^{3x^2 + x}$ find $f'(2)$.

Solution:

Differentiate: $f(x)=e^{3x^2 + x}$

$f'(x)=(6x+1)e^{3x^2 + x}$

Substitute 2 for x.

Thus, $f'(2)=13e^{14}$

2. Find the slope of the tangent to the function: $f(x)=2^{x^2+3x$ when
$x$ is equal to 3.

Solution:

Differentiate: $f(x)=2^{x^2+3x}$

$f'(x)=2^{x^2+3x} * ln2 * 2x+3$

Thus, $f'(3)=(2^{18})(ln2) * 9; f'(3)=1,635,339.371$

Sincerely,

Raymond
• Aug 24th 2011, 11:06 AM
Siron
Re: Derivatives of Exponential Functions
They're both correct!
Only 1,635,.. what do you mean with this?
• Aug 24th 2011, 11:12 AM
raymac62
Re: Derivatives of Exponential Functions
Well, I mean that's what the slope of the tangent works out to be if you actually perform the calculations I have indicated in my solution:

$f'(3)=(2^{18})(ln2)(9)$
• Aug 24th 2011, 11:14 AM
raymac62
Re: Derivatives of Exponential Functions
Is it not common practice to fully expand upon one's solution? Because I have been baffled with some large numbers as solutions which seem untypical to the course. Though the numbers, as far as I can tell are accurate. The questions above being one example.
• Aug 24th 2011, 11:17 AM
Siron
Re: Derivatives of Exponential Functions
Quote:

Originally Posted by raymac62
Well, I mean that's what the slope of the tangent works out to be if you actually perform the calculations I have indicated in my solution:

$f'(3)=(2^{18})(ln2)(9)$

I would just use this answer and don't expand it, but if you want to expand it that's offcourse no problem.
• Aug 24th 2011, 11:21 AM
raymac62
Re: Derivatives of Exponential Functions
Yeah, I guess that's what I was sort of curious about most with these solutions. I was 99% sure I had nailed the derivatives. I just wasn't sure what the mathematical conventions were in terms of presenting the final solution.

The first one involving Euler's constant seems very logical that you would leave it as is but I wasn't sure about the other one.
• Aug 24th 2011, 11:39 AM
Siron
Re: Derivatives of Exponential Functions
Yes, but I also think you can simplify further with the not expanded form if that would be necessary.
• Aug 24th 2011, 01:35 PM
HallsofIvy
Re: Derivatives of Exponential Functions
Quote:

Originally Posted by raymac62
Hi,

I was wondering if someone could check my work for the following two short problems:

1. If $f(x)=e^{3x^2 + x}$ find $f'(2)$.

Solution:

Differentiate: $f(x)=e^{3x^2 + x}$

$f'(x)=(6x+1)e^{3x^2 + x}$

Substitute 2 for x.

Thus, $f'(2)=13e^{14}$

2. Find the slope of the tangent to the function: $f(x)=2^{x^2+3x$ when
$x$ is equal to 3.

Solution:

Differentiate: $f(x)=2^{x^2+3x}$

$f'(x)=2^{x^2+3x} * ln2 * 2x+3$

Technical point: Better to use parentheses as you did with the first problem:
$2^{x^2+ 3x}(ln2)(2x+ 3)$
By "precedence of arithmetic operators", what you wrote should be interpreted as
$2^{x^2+ 3x}(ln2)(2x)+ 3$

Quote:

Thus, $f'(3)=(2^{18})(ln2) * 9; f'(3)=1,635,339.371$
In addition to what others have said, not that this is approximately correct because you have rounded of while $2^{x^2+ 3x}(ln 2)(3x+3)$ is exact. I would not got to additional work to right an answer that is only approximate when I already had the exact answer.

Quote: