# Help on logarithmic differentiation problem

• crm08
In summary, the conversation discusses using logarithmic differentiation to find the derivative of the function y = (sin(x))^(ln(x)). The steps involve taking the natural log of both sides and using the product rule to find the derivative. There is some confusion about the use of exponents and the correct trigonometric function to use, but in the end, the derivative is simplified to y * [(ln(x)) * cot(x) + (ln(sin(x))) / x].
crm08

## Homework Statement

Use logarithmic differentiation to find the derivative of the function.

## Homework Equations

y = (sin(x))^(ln(x))

## The Attempt at a Solution

I guessing the first step is raise both sides to "e", but so far I have only completed problems by taking the natural log of both sides.

(1) e^(y) = e^(sin(x))^(ln(x))

(2) d/dx (e^y) = d/dx (e^(sin(x))^(ln(x)))

(3) Can I start moving exponents in front of "e" to use the product rule, if so, do I bring "sin(x)^(ln(x)) down?, or am I on the completely wrong track?

Hi crm08!
crm08 said:
y = (sin(x))^(ln(x))

(1) e^(y) = e^(sin(x))^(ln(x))

= e^(sin(x)*ln(x))

Oh I gotcha, so now use:

d/dx (e^g(x)) = e^g(x) * g'(x), where g(x) = (sin(x))(ln(x)), and use product rule for g'(x)

I think I'm on the right track

use ln not e, since $$\ln{x^a} =a \ln{x}$$
So that $$\ln{y} = \ln( \sin{x}^{\ln{x}} ) = \ln(x) \ln(sin{x})$$

tiny-tim, I'm confused, why would $$e^y = e^{(\sin(x)^{\ln(x)})}$$ equals $$e^{(\sin(x)*\ln(x))}$$

ok I'm getting

(1/y) * dy/dx = (ln(x))*(1/sin(x))*(cos(x)) + (ln(sin(x))*(1/x)

= y*[(ln(x))*(tan(x)) + ((ln(sin(x))) / x)]

am I getting any closer?

wsalem said:
tiny-tim, I'm confused, why would $$e^y = e^{(\sin(x)^{\ln(x)})}$$ equals $$e^{(\sin(x)*\ln(x))}$$

It wouldn't. In general, (a^b)^c = a^(bc), but not a^(b^c). Take an example using a=3, b=2, c=3.

(a^b)^c = 729= a^(bc). a^(b^c) = 3^8, different.

I'm sure tiny-tim just got a bit confused.

As for crm08 - the tan should be a cot, and you should not use an equals signs when you actually multiplied by y, but other than that its looking sweet.

oops!

Gib Z said:
I'm sure tiny-tim just got a bit confused.

mmm … all those ^s

it'll be a lot easier if everyone uses the X2 tag (just above the reply box)

let's see, it should have been …

sinxlnx = (eln(sinx))lnx = eln(sinx)*lnx

ahhh, so you were referring to y and not $$e^y$$ that equals $$e^{\ln(sinx)*\ln{x}}$$.
That makes sense now!

## 1. What is logarithmic differentiation and when is it used?

Logarithmic differentiation is a method used to find derivatives of functions that are difficult to differentiate using traditional methods, such as the product or quotient rule. It is typically used when the function involves multiple products, quotients, or powers, making it easier to simplify the expression using logarithms before finding the derivative.

## 2. How do you use logarithmic differentiation to find the derivative of a function?

To use logarithmic differentiation, first take the natural logarithm of both sides of the function. Then, use properties of logarithms to simplify the expression. Next, take the derivative of both sides using the chain rule, product rule, or quotient rule as needed. Finally, solve for the original function's derivative by exponentiating both sides.

## 3. What are the benefits of using logarithmic differentiation?

Logarithmic differentiation allows for the differentiation of complex functions that cannot be easily differentiated using traditional methods. It also simplifies the process by using logarithms to handle products, quotients, and powers, making it easier to take the derivative of the original function.

## 4. Are there any limitations to using logarithmic differentiation?

Logarithmic differentiation is not always the most efficient method for finding derivatives and may not be necessary for simpler functions that can be easily differentiated using traditional methods. It also requires a good understanding of logarithmic properties and the chain rule to use effectively.

## 5. Can logarithmic differentiation be used for all types of functions?

Logarithmic differentiation can be used for most types of functions, including exponential, logarithmic, trigonometric, and algebraic functions. However, it may not be necessary or the most efficient method for simpler functions that can be differentiated using traditional methods.

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