How Does t^(n-1) Arise in Differentiating f(xt, yt) with Respect to t?

In summary, the conversation is about finding the partial derivative of a function f(xt, yt) with respect to t, where f_x denotes the partial derivative with respect to x. The answer is given as f_t * x * t^(n-1) + f_t * y * t^(n-1), and it is noted that f is a function such that f(xt, yt) = t^n * f(x, y). The speaker suggests looking in a calculus book for the chain rule for functions of several variables. They also clarify that t^(n-1) comes from the given formula for differentiation.
  • #1
JasonRox
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I'm working on this question, and I have no idea where they are getting this.

They said to differentiate f(xt,yt) with respect to t where.

Note: [itex]f_x[/itex] denotes partial derivative with respect to x.

The answer is coming up as, from the book...

[tex]f_t * x * t^{(n-1)} + f_t * y * t^{(n-1)}[/tex]

Note: If it helps, f is a function such that f(xt,yt) = t^n * f(x,y).
 
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  • #2
Try looking in your calculus book for 'chain rule for functions of several variables' or some similar topic.

It's just a differentiation formula.
 
  • #3
Yeah, that's exactly what I looked at.

Where does t^(n-1) come from?

Note: f(xt,yt)=t^n f(x,y)
 

Related to How Does t^(n-1) Arise in Differentiating f(xt, yt) with Respect to t?

What are partial derivatives?

Partial derivatives are a type of derivative that measures the rate of change of a multivariable function with respect to one of its variables while holding all other variables constant. Essentially, it is the derivative of a function with respect to one of its variables, treating all other variables as constants.

How do you find a partial derivative?

To find a partial derivative, you first need to identify the variable you are taking the derivative with respect to. Then, you simply treat all other variables as constants and take the derivative using the standard rules of differentiation.

Why are partial derivatives important?

Partial derivatives are important because they allow us to understand how a multivariable function changes in relation to each of its variables. This is crucial in many fields, including physics, economics, and engineering, where functions often involve multiple variables.

Can partial derivatives be used to find maximum or minimum points?

Yes, partial derivatives can be used to find maximum and minimum points of a multivariable function. To do this, you would take the partial derivatives with respect to each variable and set them equal to zero. The resulting system of equations can then be solved to find the critical points, which can be maximum, minimum, or saddle points.

What is the difference between partial derivatives and total derivatives?

The main difference between partial derivatives and total derivatives is that partial derivatives only consider the change in one variable at a time, while total derivatives take into account the overall change in a function with respect to all of its variables. Total derivatives are also more complex and require the use of the chain rule in addition to the standard rules of differentiation.

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