# Chain rule and partial derivatives

## Homework Statement

Suppose the differentiable function f(x,y,z) has the partial derivatives fx(1,0,1) = 4, fy(1,0,1) = 1 and fz(1,0,1) = 0. Find g'(0) if g(t) = f(t2 + 1, t2-t, t+1).

## The Attempt at a Solution

Ok I'm given the solution for this and I'm trying to work through it but I'm confused.

I understand we have g(t) = f(x(t),y(t),z(t)) where x = t2+1; y = t2-t; z = t+1.

So i thought I would just use the chain rule to find g'(t) and plug in 0 for t. But I check the solutions sheet and he uses a very different (easier) method that I don't understand. ( I just realized that we aren't given a function f in terms of x y or z so thats why this wouldn't work and we need another method. Still I don't understand this other method.)

First he says when t=0 (x,y,z)|t=0 = (1,0,0) (I think he may have made a mistake, shouldn't it be (1,0,1) ?) And then he gets a point P = (1,-1,0) No idea were this point comes from...

Next he sets up the chain rule equation: $$\frac{dg}{dt} = \frac{\partial f}{\partial x} \frac{dx}{dt} + \frac{\partial f}{\partial y} \frac{dy}{dt} + \frac{\partial f}{\partial z} \frac{dz}{dt}$$

Now I think he somehow uses that point and the partial derivatives given above to solve the equation and he gets: = 1*0 + 1*(-1) + (0*1) = -1 (depending on whether P was correct above may change the answer here I think)

So could someone please explain the method used here?

Mark44
Mentor

## Homework Statement

Suppose the differentiable function f(x,y,z) has the partial derivatives fx(1,0,1) = 4, fy(1,0,1) = 1 and fz(1,0,1) = 0. Find g'(0) if g(t) = f(t2 + 1, t2-t, t+1).

## The Attempt at a Solution

Ok I'm given the solution for this and I'm trying to work through it but I'm confused.

I understand we have g(t) = f(x(t),y(t),z(t)) where x = t2+1; y = t2-t; z = t+1.

So i thought I would just use the chain rule to find g'(t) and plug in 0 for t. But I check the solutions sheet and he uses a very different (easier) method that I don't understand. ( I just realized that we aren't given a function f in terms of x y or z so thats why this wouldn't work and we need another method. Still I don't understand this other method.)

First he says when t=0 (x,y,z)|t=0 = (1,0,0) (I think he may have made a mistake, shouldn't it be (1,0,1) ?) And then he gets a point P = (1,-1,0) No idea were this point comes from...

Next he sets up the chain rule equation: $$\frac{dg}{dt} = \frac{\partial f}{\partial x} \frac{dx}{dt} + \frac{\partial f}{\partial y} \frac{dy}{dt} + \frac{\partial f}{\partial z} \frac{dz}{dt}$$
Another way to write this is g'(t) = fx * x'(t) + fy * y'(t) + fz * z'(t), and what you want is g'(0). When t = 0, what are x(0), y(0), and z(0)? You need to evaluate all three partial deriviatives at the point (x, y, z) for which t = 0, and you need to evaluate all three ordinary derivatives at t = 0.
Now I think he somehow uses that point and the partial derivatives given above to solve the equation and he gets: = 1*0 + 1*(-1) + (0*1) = -1 (depending on whether P was correct above may change the answer here I think)

So could someone please explain the method used here?

Another way to write this is g'(t) = fx * x'(t) + fy * y'(t) + fz * z'(t), and what you want is g'(0). When t = 0, what are x(0), y(0), and z(0)? You need to evaluate all three partial deriviatives at the point (x, y, z) for which t = 0, and you need to evaluate all three ordinary derivatives at t = 0.

Ah, so then x'(0) = 2t = 2(0) = 0
y'(0) = 2t -1 = -1
z'(0) = 1

Now I understand where those come from but I'm still not sure what you mean by "You need to evaluate all three partial derivatives at the point (x, y, z) for which t = 0".

Mark44
Mentor
Ah, so then x'(0) = 2t = 2(0) = 0
y'(0) = 2t -1 = -1
z'(0) = 1
Not quite. You are confusing the derivatives of x(t), y(t), and z(t) with the value of each derivative at a specific value of t. IOW, x'(t) = 2t, but x'(0) = 0, and similarly for y'(t) vs. y'(0) and z'(t) vs. z'(0). In each case, the deriviative is a function, while the derivative evaluated at a particular value of t is just a number. x'(t) is different from x'(0), y'(t) is different from y'(0), and z'(t) is different from z'(0).

The same difference exists between g'(t) and g'(0). Using the chain rule you can write g'(t) = fx(x,y,z) * x'(t) + fy(x,y,z) * y'(t) + fz(x,y,z) * z'(t). Since you aren't given any details about f(x, y, z), it's not possible to calculate the functions fx(x, y, z), fy(x, y, z), and fz(x, y, z). You are, however given the values of these partials at a particular point (1, 0, 1) that corresponds to t = 0.

So g'(0) = fx(1,0,1) * x'(0) + fy(1,0,1) * y'(0) + fz(1,0,1) * z'(0). The only reason I have used the subscript notation (as opposed to the Leibniz notation you used) for the partial derivatives is that it's a little easier to explicitly show that they are to be evaluated at a particular point.
Now I understand where those come from but I'm still not sure what you mean by "You need to evaluate all three partial derivatives at the point (x, y, z) for which t = 0".

Not quite. You are confusing the derivatives of x(t), y(t), and z(t) with the value of each derivative at a specific value of t. IOW, x'(t) = 2t, but x'(0) = 0, and similarly for y'(t) vs. y'(0) and z'(t) vs. z'(0). In each case, the deriviative is a function, while the derivative evaluated at a particular value of t is just a number. x'(t) is different from x'(0), y'(t) is different from y'(0), and z'(t) is different from z'(0).

Isn't this what I did? I think I may have messed up my notation a bit but I have x'(t) = 2t then x'(0) = 0; y'(t) = 2t-1 then y'(0) = -1; z'(t) = 1 then z'(0) = 1;

The same difference exists between g'(t) and g'(0). Using the chain rule you can write g'(t) = fx(x,y,z) * x'(t) + fy(x,y,z) * y'(t) + fz(x,y,z) * z'(t). Since you aren't given any details about f(x, y, z), it's not possible to calculate the functions fx(x, y, z), fy(x, y, z), and fz(x, y, z). You are, however given the values of these partials at a particular point (1, 0, 1) that corresponds to t = 0.

So here since the point (1,0,1) corresponds to g(t) when t is 0 we can use the value of the partial at (1,0,1) as the value of fx(x,y,z)? Which would mean at (1,0,1) fx = 4; fy = 1; fz = 0.

g'(0) = fx(1,0,1) * x'(0) + fy(1,0,1) * y'(0) + fz(1,0,1) * z'(0). The only reason I have used the subscript notation (as opposed to the Leibniz notation you used) for the partial derivatives is that it's a little easier to explicitly show that they are to be evaluated at a particular point.

So if everything above is correct that gives me: 4*0 + 1*(-1) + 0*1 = -1.

Also, does that mean (if this is all correct) this method can only be used if the point the partials are evaluated at and the value of t are directly related as in this example, correct?

Mark44
Mentor
Isn't this what I did? I think I may have messed up my notation a bit but I have x'(t) = 2t then x'(0) = 0; y'(t) = 2t-1 then y'(0) = -1; z'(t) = 1 then z'(0) = 1;
It's not really "messing up the notation a bit." You wrote some things that just plain aren't true. Here's what you wrote that I commented on.
nhartung said:
Ah, so then x'(0) = 2t = 2(0) = 0
y'(0) = 2t -1 = -1
x'(0) $\neq$ 2t $\neq$ 0 and
y'(0) $\neq$ 2t - 1 $\neq$ -1

Since you apparently didn't understand the difference between a function of t and the value of a function at a specific number, I assumed that was the reason you were having such difficulties understanding that you needed to evaluated the partial derivatives at the point corresponding to t = 0, namely (1, 0, 1).
So here since the point (1,0,1) corresponds to g(t) when t is 0 we can use the value of the partial at (1,0,1) as the value of fx(x,y,z)? Which would mean at (1,0,1) fx = 4; fy = 1; fz = 0.
A less roundabout way to say "g(t) when t is 0" is g(0).
Here are the relationships.
x(t) = t2 + 1
y(t) = t2 - t
z(t) = t + 1

and
x(0) = 1
y(0) = 0
z(0) = 1

We don't know the formulas of any of the three partials, so for example, we can't evaluate fx(x, y, z) at an arbitrary value of t - which determines the values of x, y, and z, but we are given fx(1, 0, 1) as being 4.
Similarly, we don't know fy(x, y, z) or fz(x, y, z), but we're given the values of these partials when t = 0, which means we know the value of each partial evaluated at (1, 0, 1).
So if everything above is correct that gives me: 4*0 + 1*(-1) + 0*1 = -1.
Yes, but you should write this in context. This is g'(0) = -1.
Also, does that mean (if this is all correct) this method can only be used if the point the partials are evaluated at and the value of t are directly related as in this example, correct?
Correct. If we didn't have the information about the values of the partials for t = 0 (i.e., at (1, 0, 1)), we wouldn't be able to answer the question.

Ok. My professor made several errors in his work on the solutions sheet which made this even more confusing for me. Thanks a lot for your help.

Mark44
Mentor
Sure, you're welcome. I'm hopeful that this is less confusing for you now.