- #1
verdverm
- 7
- 0
I am having trouble finding the rule for the (partial) derivative of an expression like
y = f(X)^{g(X)}
can anyone help?
y = f(X)^{g(X)}
can anyone help?
Derivative Rule for y = f(X)^{g(X)}: Can Anyone Help?
- Thread starter verdverm
- Start date
In summary, the conversation is about finding the rule for the (partial) derivative of an expression with the form y = f(x)^{g(x)}. There is a discussion about using the hint of rewriting the expression as e^{f(x)g(x)} and agreeing on the notation. The conversation also mentions looking at Wolfram Alpha for examples of partial derivatives and discussing the meaning of the notation used.
- #2
- 22,183
- 3,321
Hint:
[tex]f(x)^{g(x)}=e^{f(x)g(x)}[/tex]
[tex]f(x)^{g(x)}=e^{f(x)g(x)}[/tex]
- #3
Mark44
Mentor
- 37,654
- 9,905
Make that eg(x) * ln(f(x)) and I'll agree.
- #4
- 22,183
- 3,321
Mark44 said:
Oh my, I should pay better attention while posting
- #5
verdverm
- 7
- 0
hmm
I was just looking at:
wolfram alpha for
[tex]
d/dx \ f(x,y)^{g(x,y)} = f^{(1,0)}(x,y)g(x,y)f(x,y)^{g(x,y)-1} + g^{(1,0)}(x,y)f(x,y)^{g(x,y)}log(f(x,y) \\
d/dy \ f(x,y)^{g(x,y) } = f^{(0,1)}(x,y)g(x,y)f(x,y)^{g(x,y)-1} + g^{(0,1)}(x,y)f(x,y)^{g(x,y)}log(f(x,y) \\
d/dx \ e^{f(x,y)g(x,y)} = f^{(1,0)}(x,y)g(x,y)e^{f(x,y)g(x,y)} + f(x,y)g^{(1,0)}(x,y)e^{f(x,y)g(x,y)}
[/tex]
and was unsure what the [tex] d/dx \ f(x,y) \ is\ f^{(1,0)or(0,1)}(x,y)[/tex] meant in the previous equations.
Is it just the partial derivative? or partial at the point (1,0) or (0,1)?
I was just looking at:
wolfram alpha for
[tex]
d/dx \ f(x,y)^{g(x,y)} = f^{(1,0)}(x,y)g(x,y)f(x,y)^{g(x,y)-1} + g^{(1,0)}(x,y)f(x,y)^{g(x,y)}log(f(x,y) \\
d/dy \ f(x,y)^{g(x,y) } = f^{(0,1)}(x,y)g(x,y)f(x,y)^{g(x,y)-1} + g^{(0,1)}(x,y)f(x,y)^{g(x,y)}log(f(x,y) \\
d/dx \ e^{f(x,y)g(x,y)} = f^{(1,0)}(x,y)g(x,y)e^{f(x,y)g(x,y)} + f(x,y)g^{(1,0)}(x,y)e^{f(x,y)g(x,y)}
[/tex]
and was unsure what the [tex] d/dx \ f(x,y) \ is\ f^{(1,0)or(0,1)}(x,y)[/tex] meant in the previous equations.
Is it just the partial derivative? or partial at the point (1,0) or (0,1)?
- #6
verdverm
- 7
- 0
my intuition is that it's a bit vector to show which variable the derivative is with respect to
Related to Derivative Rule for y = f(X)^{g(X)}: Can Anyone Help?
1. What is a partial derivative?
A partial derivative is a mathematical concept used in multivariable calculus to calculate the rate of change of a function with respect to one of its variables while holding all other variables constant. It is denoted by ∂ (partial symbol) and is often used in the study of natural sciences and engineering.
2. What are the basic rules for calculating partial derivatives?
The basic rules for calculating partial derivatives include the power rule, product rule, quotient rule, and chain rule. These rules are similar to the rules for calculating derivatives in single variable calculus, but they take into account the multiple variables involved in the function.
3. How are partial derivatives used in real-world applications?
Partial derivatives are used in a variety of real-world applications, such as in physics, economics, and engineering. They can be used to determine the rate of change of a physical quantity, optimize a system, or analyze the behavior of a function in a complex system.
4. What is the difference between a partial derivative and an ordinary derivative?
The main difference between a partial derivative and an ordinary derivative is that a partial derivative calculates the rate of change of a function with respect to one variable while holding all other variables constant, whereas an ordinary derivative calculates the rate of change of a function with respect to a single variable.
5. Are there any limitations or restrictions to using partial derivatives?
Yes, there are some limitations and restrictions to using partial derivatives. For example, the function must be differentiable, and the variables must be continuous. In some cases, the partial derivatives may also not be defined or may not exist due to the complexity of the function.
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