Find the first partial derivative of

In summary, the first partial derivatives of sin(0x+5y+z)=0 at (0,0,0) are ∂z/∂x= 0 and ∂z/∂y= -5. The equation 0x+5y+z=kπ is used to find the values of k and π that make the relation true. It simplifies to ∂z/∂x= 0 and ∂z/∂y= -5, making it easier to find the partial derivatives. The significance of k and π in this equation is that they represent the values of sin(0) and the number of full rotations around the unit circle, respectively.
  • #1
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Homework Statement



Find the first partial derivatives ∂z/∂x and ∂z/∂y of sin(0x+5y+z)=0 at (0,0,0).

Homework Equations



sin(0x+5y+z)=0


The Attempt at a Solution



0x+5y+z=kπ
z=kπ-5y

So,

∂z/∂x= 0 and ∂z/∂y= -5

What I do not understand is WHY 0x+5y+z=kπ is an acceptable equation. I found it from a solution of this online, but with no explanation. What is the significance of k and π, and why can we basically ignore the sin?
 
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  • #2
You have the relation ##\sin(\theta)=0:\theta=5y+z## right ... so what values of ##\theta## make the relation true?
 
  • #3
Simon Bridge said:
You have the relation ##\sin(\theta)=0:\theta=5y+z## right ... so what values of ##\theta## make the relation true?

Oh, duh! Sin(0)=0.

Thanks
 
  • #4
##\sin k\pi=0:k=0,1,2,\cdots##
... sometimes you are staring right at it.
 

FAQ: Find the first partial derivative of

1. What is a partial derivative?

A partial derivative is a mathematical concept used to describe the rate of change of a function with respect to one of its variables while holding all other variables constant. It allows us to analyze the behavior of a multi-variable function by looking at how it changes in one direction at a time.

2. Why is finding the first partial derivative important?

Finding the first partial derivative is important because it helps us understand how a function behaves in different directions. By finding the partial derivatives, we can determine the slope of the function in a particular direction and identify any critical points or extrema.

3. How do you find the first partial derivative?

To find the first partial derivative, we use the derivative rules for single-variable functions and treat all other variables as constants. We differentiate the function with respect to the variable we are interested in, while holding all other variables constant.

4. Can you give an example of finding the first partial derivative?

Sure, let's say we have a function f(x,y) = x^2 + 2xy + y^2. To find the first partial derivative with respect to x, we treat y as a constant and use the power rule to get fx(x,y) = 2x + 2y. Similarly, to find the first partial derivative with respect to y, we treat x as a constant and get fy(x,y) = 2x + 2y.

5. How is the first partial derivative related to the total derivative?

The first partial derivative is just one component of the total derivative. The total derivative takes into account the changes in all variables while the first partial derivative only looks at the change in one variable. The total derivative is useful for understanding how a function changes in all directions, while the first partial derivative is useful for understanding how it changes in one specific direction.

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