Derivative of a surd containing exponential sum

[basher87]

Homework Statement

y = sqrt(exp(x) - exp(-x))

Homework Equations

dy/dx = dy/du.du/dx - chain rule
d/dx(exp(x)) = exp(x) - derivative of exp(x)

The Attempt at a Solution

y = sqrt(exp(x) - exp(-x))
y' = (1/2).(1/sqrt(exp(x) - exp(-x))).[exp(x) - (exp(-x).-1)]

y' = [exp(x) + exp(-x)]/[2*sqrt(exp(x) - exp(-x))]

i've tried logarithmic differentation as well coming up with the same answer. derivative calculator says different. please help

 

[SammyS]

Homework Statement

y = sqrt(exp(x) - exp(-x))

Homework Equations

dy/dx = dy/du.du/dx - chain rule
d/dx(exp(x)) = exp(x) - derivative of exp(x)

The Attempt at a Solution

y = sqrt(exp(x) - exp(-x))
y' = (1/2).(1/sqrt(exp(x) - exp(-x))).[exp(x) - (exp(-x).-1)]

y' = [exp(x) + exp(-x)]/[2*sqrt(exp(x) - exp(-x))]

I've tried logarithmic differentiation as well coming up with the same answer. derivative calculator says different. please help

Hello basher87. Welcome to PF !

What does the derivative calculator say ? Something with cosh and tanh functions ?

 

[basher87]

thankyou sammy

no it comes up with this.

i have checked the hyperbolic functions, in fact i just substituted 2sinhx for exp(x) - exp(-x)

and got the same answer

(exp(-x)*(exp(2.5x) + exp(.5x)))/2(sqrt(exp(2x) - 1) is what the calculator gives

 

[SammyS]

thankyou sammy

no it comes up with this.

i have checked the hyperbolic functions, in fact i just substituted 2sinhx for exp(x) - exp(-x)

and got the same answer

(exp(-x)*(exp(2.5x) + exp(.5x)))/(2(sqrt(exp(2x) - 1)) is what the calculator gives

Is that possibly equivalent to your answer?

 

[basher87]

i solved the equation in terms of the hyperbolic function.

y = sqrt(2sinh x), the derivative calculator gave te same output.

if i solve it in terms of the exponential equation i get the equivalent but the calculator doesnt. Is it possible that it is a syntax error

 

[SammyS]

thankyou sammy

no it comes up with this.

i have checked the hyperbolic functions, in fact i just substituted 2sinhx for exp(x) - exp(-x)

and got the same answer

(exp(-x)*(exp(2.5x) + exp(0.5x)))/2(sqrt(exp(2x) - 1) is what the calculator gives

$\displaystyle \frac{e^{-x}(e^{2.5x} + e^{0.5x})}{2\sqrt{e^{2x} - 1}}$

$\displaystyle =\frac{e^{-x}e^{1.5x}(e^{x} + e^{-x})}{2\sqrt{e^{2x} - 1}}$

$\displaystyle =\frac{e^{x} + e^{-x}}{2e^{-0.5}\sqrt{e^{2x} - 1}}$

$\displaystyle =\frac{e^{x} + e^{-x}}{2\sqrt{e^{x} - e^{-x}}}$
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