differentiation of exponential functions question, please help (1 Viewer)

miracu · Feb 28, 2017

If f(x)= (a/2)(e^x/a + e^-x/a), find f'(a)

1729 · Feb 28, 2017

miracu said:
If f(x)= (a/2)(e^x/a + e^-x/a), find f'(a)

$$\noindent Given $f(x) = \frac{a}{2} \left(e^{x/a} + e^{-x/a}\right)$ \\ Well $\frac{\textrm{d}}{\textrm{d}x}f(x) = \frac{a}{2}\left(\frac{1}{a} \cdot e^{x/a} - \frac{1}{a} \cdot e^{-x/a}\right) = \frac{1}{2}\left(e^{x/a} - e^{-x/a}\right).$ \\ Thus, $f'(a) = \frac{1}{2}\left(e-e^{-1}\right)$

davidgoes4wce · Feb 28, 2017

$$ I assume the question to be in the form $ f(x)=\frac{a}{2}(\frac{e^x}{a}+\frac{e^{-x}}{a})$

$f'(x) =\frac{a}{2} (\frac{e^x}{a}-\frac{e^{-x}}{a})$

$f'(a)=\frac{a}{2}(\frac{e^a}{a}-\frac{1}{ae^a}$

$=\frac{1}{2}(e^a-\frac{1}{e^a})$

miracu · Feb 28, 2017

sorry, this was the given equation

Screen Shot 2017-02-28 at 10.17.08 PM.png

miracu · Feb 28, 2017

1729 said:
$$\noindent Given $f(x) = \frac{a}{2} \left(e^{x/a} + e^{-x/a}\right)$ \\ Well $\frac{\textrm{d}}{\textrm{d}x}f(x) = \frac{a}{2}\left(\frac{1}{a} \cdot e^{x/a} - \frac{1}{a} \cdot e^{-x/a}\right) = \frac{1}{2}\left(e^{x/a} - e^{-x/a}\right).$ \\ Thus, $f'(a) = \frac{1}{2}\left(e-e^{-1}\right)$

Sorry to be a hassle, but the textbook I'm using only gives the answer and shows no working out or gives any similar examples. I get how you found f'(x) but how did you find f'(a) from that point?

1729 · Feb 28, 2017

miracu said:
Sorry to be a hassle, but the textbook I'm using only gives the answer and shows no working out or gives any similar examples. I get how you found f'(x) but how did you find f'(a) from that point?

You substitute x = a into f'(x). The exponents simply become a/a = 1 and -a/a = -1

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differentiation of exponential functions question, please help (1 Viewer)

miracu

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