Differentiation of function x^x

In summary, the conversation is about differentiating a function, x^{sin3x}, using the chain rule and the correct solution is found to be dy/dx = e^{sin(3x)ln(x)}\times(sin(3x)/x + 3ln(x)cos(3x)). The OP had initially made a mistake but corrected it.
  • #1
Patjamet
6
0
Just wondering if I have done this correctly?

Homework Statement


Differentiate:

[tex]x^{sin3x}[/tex]

The Attempt at a Solution



[tex]x^{sin3x}=e^{sin(3x)ln(x)[/tex]

Employing chain rule.

[tex]y=e^{u}[/tex]
[tex]u=sin(3x)ln(x)[/tex]

[tex]dy/dx = e^{u}\times(sin(3x)/x + 3ln(x)cos(3x))[/tex]

Final Solution? =

[tex]dy/dx = e^{sin(3x)ln(x)}\times(sin(3x)/x + 3ln(x)cos(3x))[/tex]
 
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  • #2
EDIT: Made a mistake while entering your proposed answer, you're correct.
 
Last edited:
  • #4
Patjamet said:
[tex]x^{sin3x}=e^{sin(3x)ln(x)}[/tex]

The OPs result is equivalent to yours.
 
  • #5
Thanks guys.

May I ask what the "OP" is?
 
  • #6
Patjamet said:
Thanks guys.

May I ask what the "OP" is?

Original Post or Original Poster.
 

1. What is the basic concept of differentiation of function x^x?

The basic concept of differentiation of function x^x is to find the rate of change of the function at a specific point, which is known as the derivative. This is done by finding the slope of the tangent line to the curve at that point.

2. How is the derivative of x^x calculated?

The derivative of x^x is calculated using the power rule, which states that the derivative of x^n is n*x^(n-1). In the case of x^x, the derivative is x^x * (ln(x)+1).

3. What is the significance of the derivative in relation to x^x?

The derivative of x^x represents the instantaneous rate of change of the function at a specific point. This allows us to understand the behavior of the function and make predictions about its growth or decay.

4. How is the differentiation of x^x used in real life?

Differentiation of x^x is used in various fields such as physics, economics, and engineering. In physics, it is used to calculate the velocity and acceleration of an object in motion. In economics, it is used to find the marginal cost and revenue of a product. In engineering, it is used to optimize design parameters for efficiency.

5. Are there any special cases when differentiating x^x?

Yes, when differentiating x^x, we need to be careful of the cases when x is equal to 0 or 1. In these cases, the derivative becomes undefined or indeterminate, and we need to use other methods such as the limit definition of derivative to find the derivative at these points.

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