# Derivative of a surd containing exponential sum

• basher87
In summary, the conversation is about finding the derivative of the function y = sqrt(exp(x) - exp(-x)) and the attempt at solving it using logarithmic differentiation and substitution. The person encountered a discrepancy between their answer and the answer from a derivative calculator, which prompted them to ask for help. They also mention trying to solve the equation in terms of hyperbolic functions and getting the same answer from the calculator.
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))]

basher87 said:

## 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))]

Hello basher87. Welcome to PF !

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

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)

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

basher87 said:
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)

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

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

basher87 said:
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)

(exp(-x)*(exp(2.5x) + exp(0.5x)))/2(sqrt(exp(2x) - 1) is what the calculator gives
$\displaystyle \frac{e^{-x}(e^{2.5x} + e^{0.5x})}{2\sqrt{e^{2x} - 1}}$

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

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

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

## What is a derivative of a surd containing exponential sum?

A derivative of a surd containing exponential sum refers to the rate of change of the function that is represented by the surd, with respect to the variable in the exponential term.

## How do I find the derivative of a surd containing exponential sum?

To find the derivative, you can use the power rule or chain rule, depending on the form of the surd. You can also use logarithmic differentiation to simplify the process.

## Why is it important to find the derivative of a surd containing exponential sum?

The derivative helps us to understand the behavior of the function, such as its maximum and minimum points, rate of change, and concavity. It also allows us to solve optimization problems and make predictions about the function.

## Can I use the quotient rule to find the derivative of a surd containing exponential sum?

Yes, you can use the quotient rule if the surd is in the form of a fraction. However, it may be easier to use logarithmic differentiation for more complex surds.

## Are there any real-world applications of the derivative of a surd containing exponential sum?

Yes, the concept of finding the derivative is used in various fields such as physics, engineering, economics, and finance to model and analyze real-world phenomena. It is particularly useful in predicting and optimizing the behavior of exponential growth or decay processes.

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