Differerntiating Exponential Functions

In summary, the conversation is about differentiating the function y=(\ln x)^\cos x using logarithmic differentiation. The formula for this is given as \frac{dy}{dx} = y\ln a = a^x \ln a. The conversation also touches on the Leibniz rule of differentiation and the use of complex numbers for x<0. The question asker also mentions a bonus question on their exam being different from the topic at hand.
  • #1
evan4888
11
0
I really really need help with this one. This was a bonus question for one of my previous exams. I have no idea how to work through it.

Differentiate the function:

[tex]y= ( \ln x ) ^\cos x[/tex]
 
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  • #2
Do you know about logarithmic differentiation...?If you don't,are you willing to learn? :smile:

Daniel.
 
  • #3
Logarithmic differentiation:

[tex]y = a^x[/tex]

[tex]\ln y = \ln (a^x) = x\ln a[/tex]

Then by the chain rule: [tex]\frac{1}{y} \frac{dy}{dx} = \ln a[/tex] (only true if a is a constant. If a = a(x), then you need to apply the chain and product rules to the RHS).

So, [tex]\frac{dy}{dx} = y\ln a = a^x \ln a[/tex]
 
  • #4
Is this even close to the right answer? Or do I need to also multiply by the derivative of the exponent?


[tex] ( \ln x )^\cos x [/tex] [tex] ( \ln ( \ln x) ) [/tex]
 
  • #5
I misses a few terms.You have to use the Leibniz rule of differetiantion of products of functions.

Daniel.
 
  • #6
Nylex, question asked is the differential of

[tex]y=(\ln{x})^\cos{x}[/tex]

not

[tex]y=\ln{\left (x^\cos{x} \right )}[/tex]
 
  • #7
He didn't go for the function in the OP.He exemplified what is meant by logarithmic differentiation.

Daniel.
 
  • #8
True, but it was implied that you can use the [itex]\ln{\left (x^a\right )}=a\ln{x}[/itex] rule directly for the problem at hand.
 
  • #9
In all The usual formulas elementary calculus students are expected to learn why is
(u^v)'=v*(u^(v-1))*v'+(u^v)*log(u)*v'
Rarely included. It is quite nice and it is easy to see using the chain rule.
Or Individual function differential operators.
D(u^v)=(Du+Dv)(u^v)=(Du)(u^v)+(Dv)(u^v)
where (Du)f(u,v)=Dx f(u(x),v(y))|y=x
(Dv)f(u,v)=Dx f(u(y),v(x))|y=x
Dx y=0
 
  • #10
evan4888 said:
I really really need help with this one. This was a bonus question for one of my previous exams. I have no idea how to work through it.

Differentiate the function:

[tex]y= ( \ln x ) ^\cos x[/tex]

If this is the bonus on the exams... your teacher is really nice..

my bonus question on the test was
Limit(x^x, x, 0)...
 
  • #11
leon1127 said:
If this is the bonus on the exams... your teacher is really nice..

my bonus question on the test was
Limit(x^x, x, 0)...
These should not be bonus questions.
x^x:=exp(x log(x))
x^x~1+x log(x)~1+|x|
Also le Hopitals rule works nicely.
What I want to know is did you mean the directed limit? You need complex numbers to consider x^x for x<0.
 

1. What is an exponential function?

An exponential function is a mathematical function in which the independent variable, or input, appears in the exponent. This leads to a rapid increase or decrease in the output values as the input values increase or decrease.

2. How do you differentiate an exponential function?

The process of differentiating an exponential function involves taking the derivative, or the rate of change, of the function. This can be done by using the power rule, where the exponent becomes the coefficient and the new exponent is one less than the original exponent.

3. What is the purpose of differentiating exponential functions?

Differentiating exponential functions allows us to find the slope of the function at any given point, which can be useful in solving real-world problems such as population growth or compound interest.

4. Can all exponential functions be differentiated?

Yes, all exponential functions can be differentiated as long as they do not contain any other operations, such as addition or multiplication, within the exponent.

5. Are there any special rules for differentiating exponential functions?

Yes, there are a few special rules for differentiating exponential functions, such as the natural logarithm rule and the chain rule. These rules can be used to differentiate more complex exponential functions.

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