1 year, 7 months ago
How does one use derivatives in three-dimensional graphs?
I understand doing derivatives on a Cartesian plane, but I'm curious about higher dimensional space.
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The notion of a derivative extends pretty easily to higher dimensional space. Remember the basic formula for a derivative:
df/dx = lim (f(a+h)-f(a))/h
in Cartesian space, a is a scalar. But it can also be a vector. A function in three-space can be defined by f((x,y)), where (x,y) is a two-dimensional vector. The same math applies as before. You can even generalize this to higher-order spaces.
You can also set it up so that f returns a vector, so you can have (y,z)=f(x), where (y,z) is a two-vector giving both the y and z points for any x.
Often, what you want is the _partial_ derivative; that is, the derivative with respect to one particular dimension. You might have a function z=f(x,y), and want to know how z changes with respect to x or z changes with respect to y. Here, you use a slight change in the definition of a derivative:
∂f(x,y)/∂x = lim (f(x+h, y) - f(x,y))/h
The ∂ is the "partial derivative" symbol. The formula, as you can see, is very similar to the full derivative, except that it's applying the tiny delta to only one variable. The same rules for derivatives apply, except that you treat the other variable as a constant. So, if f(x,y) = x^2y^3, ∂f/∂x = 2xy^3 (treating t^3 as a constant and taking the ordinary derivative of x), and ∂f/∂y = 3x^2y^2 (treating x^2 as constant and taking the ordinary derivative of y.
df/dx = lim (f(a+h)-f(a))/h
in Cartesian space, a is a scalar. But it can also be a vector. A function in three-space can be defined by f((x,y)), where (x,y) is a two-dimensional vector. The same math applies as before. You can even generalize this to higher-order spaces.
You can also set it up so that f returns a vector, so you can have (y,z)=f(x), where (y,z) is a two-vector giving both the y and z points for any x.
Often, what you want is the _partial_ derivative; that is, the derivative with respect to one particular dimension. You might have a function z=f(x,y), and want to know how z changes with respect to x or z changes with respect to y. Here, you use a slight change in the definition of a derivative:
∂f(x,y)/∂x = lim (f(x+h, y) - f(x,y))/h
The ∂ is the "partial derivative" symbol. The formula, as you can see, is very similar to the full derivative, except that it's applying the tiny delta to only one variable. The same rules for derivatives apply, except that you treat the other variable as a constant. So, if f(x,y) = x^2y^3, ∂f/∂x = 2xy^3 (treating t^3 as a constant and taking the ordinary derivative of x), and ∂f/∂y = 3x^2y^2 (treating x^2 as constant and taking the ordinary derivative of y.
You can leave an optional "tip" with Mahalo's virtual currency, Mahalo Dollars. If you are asking a difficult question that might require some research, or if you'd like a wide variety of feedback, a higher tip often leads to more answers to your question.
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