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ninjaduck
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Member warned about posting with no effort
Homework Statement
Find the differential of axb
Homework Equations
The Attempt at a Solution
Really not sure where to start, honestly. Thanks in advance :)
Show some effort at understanding the problem.ninjaduck said:Homework Statement
Find the differential of axb
Homework Equations
The Attempt at a Solution
Really not sure where to start, honestly. Thanks in advance :)
That's pretty difficult to read.ninjaduck said:In the picture is what I have so far.
Not sure if correct or not.
SammyS said:That's pretty difficult to read.
It looks like you let ##\displaystyle\ y=a^{\displaystyle x^b}\,,\ ## then found the derivative, ##\displaystyle\ \frac{dy}{dx}\ ##.
So, now to get the differential.
It seems correct and I have a suggestion that you can differentiate on ##x## instead of ##y## while the process is similar (just a little easy).ninjaduck said:In the picture is what I have so far.
Not sure if correct or not.
Yes. That's an excellent point!tommyxu3 said:As I also thought they are similar (derivative and differential), or maybe I'm wrong...?
It seems correct and I have a suggestion that you can differentiate on ##x## instead of ##y## while the process is similar (just a little easy).
The differential of a^x^b is given by the formula dy/dx = a^x^b * ln(a) * b.
The differential of a^x^b is derived using the chain rule and the power rule. First, we rewrite the function as (a^x)^b and then apply the chain rule to get dy/dx = b * (a^x)^(b-1) * (a^x)' = a^x^b * ln(a) * b.
Yes, the differential of a^x^b can be simplified to dy/dx = a^x^b * ln(a) * b = b * a^x^b-1.
Yes, both the values of a and b affect the differential of a^x^b. The value of a affects the overall scaling of the function, while the value of b affects the rate of change of the function.
The differential of a^x^b is used in many real-world applications, such as in finance, physics, and biology. It can be used to model exponential growth or decay, as well as various processes that involve continuously changing quantities.